Research ArticleProbabilistic Learning Coherent Point Drift for 3D UltrasoundFetal Head Registration

Jorge PerezndashGonzalez 12 Fernando Arambula Cosıo 1 Joel C Huegel 23

and Veronica Medina-Bantildeuelos 4

1Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas Universidad Nacional Autonoma de Mexico MeridaYucatan Mexico2Biomechatronics Laboratory School of Engineering and Science Tecnologico de Monterrey Guadalajara Mexico3Center for Extreme Bionics Massachusetts Institute of Technology Cambridge MA USA4Neuroimaging Laboratory Electrical Engineering Department Universidad Autonoma Metropolitana Iztapalapa Mexico

Correspondence should be addressed to Jorge PerezndashGonzalez jorgepereziimasunammx

Received 19 August 2019 Accepted 4 December 2019 Published 31 January 2020

Academic Editor Po-Hsiang Tsui

Copyright copy 2020 Jorge PerezndashGonzalez et al -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Quantification of brain growth is crucial for the assessment of fetal well being for which ultrasound (US) images are the chosenclinical modality However they present artefacts such as acoustic occlusion especially after the 18th gestational week whencranial calcification appears Fetal US volume registration is useful in one or all of the following cases to monitor the evolution offetometry indicators to segment different structures using a fetal brain atlas and to align and combine multiple fetal brainacquisitions -is paper presents a new approach for automatic registration of real 3D US fetal brain volumes volumes thatcontain a considerable degree of occlusion artefacts noise and missing data To achieve this a novel variant of the coherent pointdrift method is proposed -is work employs supervised learning to segment and conform a point cloud automatically and toestimate their subsequent weight factors -ese factors are obtained by a random forest-based classification and are used toappropriately assign nonuniform membership probability values of a Gaussian mixture model -ese characteristics allow for theautomatic registration of 3D US fetal brain volumes with occlusions and multiplicative noise without needing an initial pointcloud Compared to other intensity and geometry-based algorithms the proposed method achieves an error reduction of 74 to607 with a target registration error of only 638plusmn 324mm -is makes the herein proposed approach highly suitable for 3Dautomatic registration of fetal head US volumes an approach which can be useful to monitor fetal growth segment several brainstructures or even compound multiple acquisitions taken from different projections

1 Introduction

Fetal ultrasound (US) is the most commonly used imagingmodality in obstetrics because it does not require ionizingradiation works in real time the transducer is easilymanipulated and is inexpensive compared to other im-aging systems such as Computed Tomography (CT) orMagnetic Resonance Imaging (MRI) It provides valuableinformation of the fetal central nervous system whichpresents some of the most relevant structures for fetalclinical assessment [1] However US images present some

drawbacks such as speckle noise intensity variations de-pendent on the patient and operator and acoustic shadows-e phenomenon of acoustic occlusion occurs mainly inthe second and third trimesters of pregnancy due tononuniform fetal cranial calcification which affects theacoustic beam penetration of ultrasound waves it mayimpact on the quality of fetal brain images because they canpresent acoustic shadows and missing tissue information[1] -is phenomenon can cause difficulty in adequatemeasurement of several important brain structures such ascerebellum or nuchal translucency which are used for the

HindawiComputational and Mathematical Methods in MedicineVolume 2020 Article ID 4271519 14 pageshttpsdoiorg10115520204271519

detection of several maternal-fetal diseases or for weightestimation at birth

According to the International Society of Ultrasound inObstetrics and Gynecology [2] there are different acquisi-tion planes depending on the fetal brain structures to beanalyzed the main being axial sagittal and coronal Eachplane is obtained by changing the acquisition angle and canprovide information about brain tissue that may be com-plementary due to the phenomenon of nonuniform acousticocclusion in the fetal head Currently in clinical practiceeach plane is used individually depending on the area to beanalyzed which could be a limiting factor in the mea-surement of brain indices and in clinical assessment In thiswork a new automatic rigid registration method of several3D fetal brain views is proposed -is can be useful tocompare fetal growth in different stages of pregnancy as atool to segment different structures or to combine fetal braininformation from different views in order to attend acousticocclusion artifacts [3 4] -e method presented focuses onmodeling the fetal cranium as a point cloud and performingan automatic alignment using a Coherent Point Drift (CPD)approach weighted by the membership probability of eachpoint obtained a priori with a random forest classifier

-e alignment or registration of a set of volumes is amathematical approach that consists of finding an optimalgeometric transformation T which maps each differentvolume to a common reference view Most of the methods ofregistration are based on intensity or geometrical features[5] -e former seeks to match the intensity levels of twovolumes by a similarity measure Among the main metricsare mutual information (MI) cross correlation (CC) andmean square error (MSE) that quantify the joint infor-mation between the intensity levels of the two volumes to bealigned [6ndash9] However these algorithms can be compu-tationally expensive and are sensitive to intensity variationsor missing data

On the other hand geometric or point set corre-spondence registration methods are based on automati-cally or semiautomatically aligning distinctive features(contours intersections or corners) of an image or vol-ume Like in intensity-based methods the problem is tofind an optimal transformation matrix which registers adisplaced volume to the same coordinate space of a ref-erence volume by maximizing the correspondence be-tween a set of points or salient features that describe bothvolumes -e point cloud registration approaches haveseveral applications in computer vision pattern recog-nition object detection pose estimation medical imageanalysis modeling and feature extraction [10] -esealgorithms present several advantages they have lowercomputational cost compared to intensity-based methodsthey can be minimally sensitive to missing data or outliersand because they do not consider information intensitythey are immune to additive or multiplicative noise whichare common artefacts in US fetal images However thesealignment approaches require a point cloud that describesthe enveloping area of the volumes to register or an op-timal detection of salient features According to Savvaet al [11] intensity-based registration methods are

computationally more expensive compared to geometry-based methods however the latter show better perfor-mance when registering key points point clouds or en-velope surfaces

Several point set registration methods have been re-ported such as iterative closest point (ICP) [12 13] which isbased on establishing correspondence of pairs of points byiteratively minimizing the mean square error of the distancebetween them Random Sample Consensus (RANSAC) [14]which is based on the iterative adjustment of a model orScale-Invariant Feature Transform (SIFT) that uses multipleGaussian filters to find the salient points needed in thealignment process [15] However to perform a properalignment these algorithms require well-defined outlineslines or salient points and characteristics that are notpresent in US fetal images

In contrast Coherent Point Drift (CPD) proposed byMyronenko and Song [16] is a robust computationally fastalgorithm that achieves good results using point clouds withoutliers noise and missing data and outperforms moststate-of-the-art geometric-based approaches -ese char-acteristics make CPD a suitable method to register sets ofpoints extracted from US fetal brain volumes because asmentioned before cerebral acoustic occlusion and specklenoise in US images are crucial factors to be considered tocarry out an adequate alignment

Given a set of points to be aligned and a second pointcloud modeled as a set of centroids of a Gaussian mixturemodel (GMM) the CPD method registers both pointclouds by iteratively searching for an optimal transfor-mation matrix T Some of the most notorious disadvan-tages of the CPD method are that an equiprobablemembership of GMM is considered which can affect theregistration process because not all points contributeequally In addition it considers an isotropic covariancematrix of the GMM to ensure that the algorithm convergesFinally it incorporates a user-defined weight factor w onthe GMM to deal with outliers noise and missing data Forthese reasons several researchers have proposed variantson the CPD method

Wang et al [17] propose a search method that combinesgenetic algorithms with NelderndashMead simplex approach toautomatically define the weight factor w-eir results show areduction of up to 50 error in cases with a high number ofoutliers Gao et al [18] designed a method to update theweight factor w while the CPD algorithm iteratively alignstwo sets of points -is is achieved by incorporating thefactor w in the transformation parameters optimization-eir results show robustness against noise and outliers Liuet al [19] reported an Automatic Outlier Suppression (AOS)algorithm for the CPDmethod It is inspired by bidirectionalnormalizations of the matrix and outlier rejection in theprobabilistic relaxation labeling algorithm which can pro-vide robust point matching by preserving local neighbor-hood structures de Sousa and Kropatsch [20] implementeda variant of CPD by integrating centrality measures such asdegree betweenness closeness eigenvector and pageRankcentralities using Delaunay triangulation -ese metrics areindicators of the topological relationships between

2 Computational and Mathematical Methods in Medicine

neighboring points and were incorporated independentlyinto a probabilistic cost function -e results show im-provements in the CPD method however it is necessary tocombine these indices to achieve better performance Zhanget al [21] incorporated SURF as a preliminary step to aligncontours afterwards the membership probability of theGMM is estimated by the edge point confidence -is CPDvariant was applied to synthetic aperture radar imagesobtaining better results compared to the original method Luet al [22] proposed the accelerated CPD for fast registrationof 3D point clouds -ese authors incorporated globalsquared iterative expectation-maximization algorithm withfast Gauss transform to the CPD method managing toreduce the processing time up to ten times Other authorshave reported CPD variants seeking to optimally assign theGMM membership probability using rotation invariantshape descriptors [23] correspondence priors and corre-spondence preserving subsampling approaches [24] and theshape context of one point with respect to the distribution ofother points [25] On the other hand Saval-Calvo et al [26]proposed color-CPD algorithm to register 3D points byusing color and shape spaces to jointly estimate the bestmatch -ese authors consider that the incorporation ofcolor helps to register sets of points with noise or missingdata -e color-CPD combines the color information in theiterative process of aligning two sets of points However thismethod considers an isotropic covariance matrix and anequal membership probability in the GMM In general thesemethods report variations of the CPD algorithm seeking toimprove the computational speed the automatic weightingof the membership probability and the weight factor defi-nition to deal with outliers and missing data for differentapplications of 2D and 3D image registration A disad-vantage to highlight is that these methods need an initialpoint cloud prior to registration which is not available in USfetal brain studies because they are constituted by intensityvoxels

For the specific application of 3D US fetal head align-ment several works have been published -is is a difficulttask due tomultiplicative noise fetal movement and acousticshadows and factors that hinder an adequate alignment Inthis context Cen et al [27 28] have worked on the regis-tration of volumes using shape and texture patternsextracted from a Gabor filter bank -is set of features wasused to register studies in 2D and 3D using only one fetalbrain Fathima et al [29] reported a method based onrepresenting each image by amplitude orientation and localphase It is used to perform the affine registration based onnormalized mutual information Chen et al [30] proposedan algorithm for fetal head registration of US phantomvolumes It is based on previously matching segmentedfeatures such as the eyes or the fetal head using feature-basedregistration

In other image modalities Kuklisova et al [31] reporteda registration method between fetal brain MRI and USstudies in which the former was converted into a pseu-dovolume of US using a probabilistic atlas Subsequently thepseudovolume is aligned towards a conventional US fetalbrain study using a robust block-rigid registration -e aim

of this work was to use the information from the registeredMRI study to improve the visualization of fetal brain ana-tomical structures All these works seek to register fetalbrain studies of US-US and US-MRI in two and three di-mensions However some of them use synthetic images orphantoms Additionally in none of the cases do they con-sider studies with artefacts of fetal brain occlusion whichconsiderably affects the registration methods based on in-tensity or texture

An exhaustive review of US image registration waspublished by [32] -e authors include image-basedmethods such as MI CC or MSE correspondence-basedmethods where characteristics were selected either manu-ally or using algorithms like SIFT or other similaritymeasures such as Hellinger distance statistics-based fuzzylocal binary patterns or the sum of absolute differencesAccording to this review the most important drawbacks forUS registration are multiplicative speckle and occlusionartefacts while the main reported medical imaging appli-cations are in head neck breast heart liver kidney boneprostate and fetal studies

To the best of our knowledge there are not any researchstudies that address the problem of automatic rigid regis-tration of 3D US fetal heads based on point clouds that canwork correctly in the presence of artefacts such as multi-plicative noise occluded brain regions and outliers

In the present work a registration strategy is proposedthat has two main contributions

(1) Automatic registration of real 3D US fetal brainvolumes contains a considerable degree of occlusionartefacts noise and missing data Previous researchstudies have not addressed this issue in a real clinicalcontext As already mentioned the alignment of fetalstudies can be useful in several relevant clinicalapplications such as brain structures segmentationfetal growth monitoring multiprojection fusion oracoustic shadows mitigation

(2) A variation of the CPD method that incorporatesnew features automatic segmentation and confor-mation of the point cloud qualities that have notbeen reported by any previously developed CPDvariant together with an appropriate proposedweighting of membership probabilities of the GMMSegmentation as well as weighting factors are ob-tained from an RF-based classifier fed by a set offeatures composed of intensity texture and edgeparameters Preliminary results of this research havebeen reported in [33] in which a variant of the CPDmethod using another scheme of weighting waspresented

In Section 2 the registration methodology as well as thecharacteristics of the fetal studies used for training andvalidation is described in detail In Section 3 the obtainedresults the corresponding discussion and the comparisonagainst other intensity-based and geometric-based regis-tration methods are exposed Finally in Section 4 theconclusions drawn from this work are presented

Computational and Mathematical Methods in Medicine 3

2 Material and Method

-e overall diagram of the proposed methodology is shownin Figure 1 and will be described in detail in the followingsections It starts with the acquisition of multiple US vol-umes of the fetal head observed at different projections(Figure 1(a) Section 21) followed by a preprocessing stepwhere confidence maps and texture features are computed(Figure 1(b)) -ese parameters are used for segmentationpurposes and to estimate probabilistic weights by means ofRandom Forest classification (Figure 1(c) Section 22) -epoint clouds built from the segmented fetal head are thenweighted with the posterior probability values estimatedfrom the Random Forest process (Figure 1(d) Section 22) tofinally carry out the registration between the two pointclouds using the proposed method (Figure 1(e)) as de-scribed in Section 23

21 Fetal Brain Data Set Eighteen fetal brains between 20and 24 weeks of gestational age were acquired using acurvilinear ultrasound transducer in mode B at frequenciesof 8ndash20MHz (Voluson E8 General Electric HealthcareCompany USA) All volumes had isotropic resolutionsgoing from 02 to 05mm3 Each case consisted on twostudies taken at coronal and axial projections to the samepatient in equal conditions All studies were acquired byobstetrics experts in Fetal Medicine and were approved bythe Department of Fetal Medicine of the National Instituteof Perinatology (INPer) All patients gave their consentaccording to the Declaration of Helsinki Finally each pair ofvolumes was manually aligned by experts in obstetrics fromINPer for this several fiduciary points were considered suchas middle line peduncles and thalamus among others -isstep will be important for the evaluation and comparison ofthe proposed registration method

22RandomForestProbabilisticWeights As a preprocessingstep the confidence map was calculated for each US ac-quisition -e purpose is to emphasize shaded or attenuatedregions in US data obtaining images with homogeneousintensities It has been proven that these maps can help in theprocessing and registration of US data [34]

Subsequently a binary classification scheme based onRandom Forest (RF) which is a machine learning algorithmbased on the ensemble of multiple decision trees [35] wasimplemented to find the weighting factors In a previousresearch a Support Vector Machine (SVM) classifier wastested for this purpose [33] with similar performance butwith a higher computational cost

Given a training data set V (x1 x2 xd) isin Rd be-longing to two classes c +1 c minus 1 where V representseach voxel xd are the responses of a chosen filter bank at thatparticular voxel and d is the feature number or extractedpatterns in each voxel -e task is to optimize an energyfunction through the training set S0 (root node) associatedwith a labeled data set -e optimization function can bedefined as follows

βj argmaxβisinψj

I Sj θ1113872 1113873 (1)

where for each subset of input training data Sj RF learns thefunction that best splits Sj into left SL

j and right SRj nodes In

this context j represents each division node β is a set of splitparameters and ψj is a random subset of the entire pa-rameter space ψ -ese parameters help to reduce possibleoverfitting thus improving the generalization capabilities ofRF For classification purposes the function I can be definedas the gain information for discrete distributions

I Sj β1113872 1113873 H Sj1113872 1113873 minus 1113944iisin LR

Sij

11138681113868111386811138681113868

11138681113868111386811138681113868

Sj

11138681113868111386811138681113868

11138681113868111386811138681113868H S

ij1113872 1113873 (2)

with i indexing the two child nodes (left L or right R node)and H(Si

j) being the entropy of a given subset Sij which can

be defined as follows

H Sij1113872 1113873 minus 1113944

cisinplusmn1P(c)logP(c) (3)

where p(c) is calculated as the normalized empirical his-togram of labels corresponding to the training points in Si

jTo assemble the result of all the decision trees given the classp(c +1 |V) the content result of each leaf is accumulatedby

P(c +1 |V) 1T

1113944

T

t1Pt(c +1 |V) (4)

where t 1 2 T represents the number of trees used inthe classification and P(c +1 |V) is denoted as the posteriorprobability of the classification process For this work the fetalcranium is considered as class c +1 (segmentation requiredto build the point cloud) and the rest of brain tissue andacoustic artefacts as class as it can be seen in Figure 2

-e task is to automatically classify these two classesusing a set of features that includes intensity informationand several texture filters such as variance rank entropymedian and Wiener estimated on an (9 times 9 times 9) analysiswindow [36] Other incorporated features were the edgesobtained from Canny [37] and Laplacian filtering Followingthe fetal cranium segmentation the point cloud is con-structed using constrained Delaunay tetrahedralization [38]-e point clouds are the vertices in cartesian coordinates ofthe obtained triangular mesh and correspond to the innerand outer enveloping areas of the fetal cranium

23 Probabilistic Learning Coherent Point Drift -e methodproposed for fetal brain volume registration is based on thealgorithm developed byMyronenko and Song [16] known asCoherent Point Drift It consists of a geometric registrationthat establishes point to point correspondences and assumesthat each element drifts coherently as a group while pre-serving the pointsrsquo cluster topology Let X and Y be two setsof points to be registered -ey can be denoted asY ym | 1 2 M1113864 1113865 and X xn | 1 2 N1113864 1113865 where Ycorresponds to the set of centroids of the GMM and X is the

4 Computational and Mathematical Methods in Medicine

generated pointsrsquo set -e task is to obtain an affine trans-form T such that X T(YΘ) where Θ is a matrixcontaining rotation translation and scaling parameters-eprobability density function (PDF) on a GMM can bewritten as follows

p(x) 1113944M+1

m1P(m)p(x | m) (5)

where

p(x | m) 1

2πσ2exp minus

x minus T ym θ( 11138571113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138682

2σ2⎧⎨

⎩

⎫⎬

⎭ m 1 2 M

(6)

and P(m) (1M) is an equiprobable distributionA uniform distribution denoted by p(x | M + 1) 1N

is used to deal with outliers occlusion artefacts and noise Aweight factor (0ltwlt 1) representing the outliersrsquo ratio isincorporated into the distribution together with an isotropicvariance σ2 term -erefore the PDF is formulated asfollows

p(x) ω1N

+(1 minus ω) 1113944M+1

m1

1M

p(x | m) (7)

Parametersrsquo matrix Θ is derived from the GMM cen-troids and can be estimated by probability maximization orby minimization of the negative logarithmic probability ofthe following function

E Θ σ21113872 1113873 minus 1113944

N

n1log 1113944

M+1

m1P(m)p xn

1113868111386811138681113868 m1113872 1113873 (8)

previously assuming that each gaussian component isindependent

Given two points sets (ym and xn) the GMM posteriorprobability is denoted by p(m | xn) P(m)p(xn | m)p(xn)-e CPD method applies an Expectation-Maximization(EM) algorithm to adjust Θ and σ2 using Bayesrsquo theorem-e posterior probabilities of the mixture componentsPold(m | xn) (equation (9)) are iteratively computed in whatis known as the expectation or E-step

Pold

m | xn( 1113857 exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967

1113936Mk1exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M (9)

tree T

tree T

3D US fetalhead acquisitions

Preprocessing andtexture

maps analysis

Fetal head segmentationand weight estimation

by random forest

Weighted pointclouds of the

fetal head

Fetal headsaligned by PL-CPD

(a) (b) (c) (d) (e)

Figure 1 Methodological diagram

250

200

150

100

50

0c = +1

c = ndash1Grayscale

Figure 2 Classes considered for fetal cranium segmentation

Computational and Mathematical Methods in Medicine 5

-e ldquonewrdquo parameter values are estimated byminimizingexpectation of the complete negative log-likelihoodfunction

Q minus 1113944N

n11113944

M+1

m1Pold

m | xn( 1113857log Pnew

(m)pnew

xn

1113868111386811138681113868 m1113872 11138731113960 1113961

(10)

during the maximization or M-step -e new Θ and σ2parameters are obtained by rewriting equation (10)

Q Θ σ21113872 1113873 12σ2

1113944

N

n11113944

M+1

m1Pold

m | xn( 1113857 xn minus T ym θ( 1113857

2

+DNp

2log σ2

(11)

where D corresponds to data dimensionality -e algorithmalternatively iterates E-step and M-step until Qrsquos conver-gence thus aligning the sets of points with X T(YΘ)

However assigning the same probability P(m) (1M)

to all GMM components can affect registration quality given

that each Gaussian contributes differently in the model Inthis work it is proposed to weight the membership prob-abilities with the values obtained from the posterior prob-ability resulting from the RF classification process At thesame time the fetal cranium classification helps to obtain thepoint clouds used in the registration

In this way the GMM membership probabilities P(m)

correspond to the posterior probability (normalized between0 and 1) obtained during the classification stepP(c +1 | xn) (equation (4)) (where c +1 is the classcorresponding to fetal cranium) for each point xn It can beexpressed by

P(m)lowast

P c +1 | xn( 1113857

λ (12)

where

λ 1113944M

k1P c +1 | xn( 1113857k (13)

-erefore Pold(m | xn) and the Q objective function canbe rewritten as follows

Poldlowast

m | xn( 1113857 P c +1 | xn( 1113857exp minus xn minus T ym θ( 1113857

22σ21113874 11138751113882 1113883

1113936Mk1P c +1 | xn( 1113857k exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M

Qlowast

Q + 1113944N

n11113944

M

m1Pold

m | xn( 1113857logP(m)lowast

(14)

With this weighting strategy outliers will have less contributioncompared to voxels that describe the fetal cranium which willhave a larger weight during the registration process -esechanges do not affect the previously described EM optimi-zation procedure -us the Probabilistic Learning CoherentPoint Drift (PL-CPD) method has the advantages of being lesssensitive to outliers and do not requiring a high computationalcost compared to other intensity-based methods

24 Evaluation of the PL-CPD Method To assess RFrsquos per-formance in head segmentation a 5-fold crossvalidation wascarried out where the training and test data sets were ob-tained from a sample of five US volumes (an example of theclasses is presented in Figure 2) As segmentation perfor-mance metric the Dice index was computed which is de-fined as follows

Dice 2|AcapB|

|A| +|B| (15)

where A is the volume segmented by an expert in obstetricsand B represents the automatically segmented volumeAdditionally the Hausdorff Surface Distance (HSD) [39]and the area under the receiver operating characteristic

(ROC) curve (AUC) were obtained After segmentationvalidation the RF was applied to the remaining studies

Subsequently to validate the proposed PL-CPDmethodthe Target Registration Error (TRE) (equation (16)) betweenthe reference volume VR and the study to be aligned VM

were measured considering the eighteen pairs of registeredvolumes A total of n 1000 uniformly distributed targetpoints were defined as ai and bi corresponding to theregistered and reference volumes respectively

TRE VR VM( 1113857

1n

1113944

n

i1bi minus ai( 1113857

2

11139741113972

(16)

According to Fitzpatrick [40] TRE provides a betterestimation of registration errors than a landmark-basedmetric Additionally to evaluate in detail translation androtation errors the Root Mean Square (RMS) difference wasused It can be denoted as follows

RMS VR VM( 1113857

1m

1113944

m

k1bk minus ak( 1113857

2

11139741113972

(17)

To measure displacement errors the values in the threeaxes (ΔX ΔY and ΔZ) are considered as bk and ak with

6 Computational and Mathematical Methods in Medicine

m 3 on the other hand for the rotation errors the values ofthe angles (α δ and λ) are taken as bk and ak

To evaluate its performance the proposed PL-CPD al-gorithm was compared with other previously reported in-tensity-based registration methods Mutual Information(MI) Cross Correlation (CC) and Mean Squared Error(MSE) [6ndash8] -e method was also compared with somegeometric registration methods based on correspondencesuch as Iterative Closest Point (ICP) [12 13] RandomSample Consensus (RANSAC) [14] Scale-Invariant FeatureTransform (SIFT) [15] and Coherent Point Drift (CPD)[16] For all comparisons TRErsquos mean and standard devi-ation were obtained from the accumulated errors measuredfor each registration case Statistical differences for thesetests were determined by an paired Student t-test (plt 005)comparing the PL-CPD method with each other In addi-tion the mean RMS errors for translations and rotationswere obtained for each carried out registration

3 Results and Discussion

In this section results of the proposed registration method(PL-CPD) as well as the corresponding discussion arepresented As mentioned in Section 22 the procedure be-gins with a preprocessing step of background normalizationwith confidence maps An example of the results obtained isshown in Figure 3 fetal head image in axial view(Figure 3(a)) its corresponding confidence map(Figure 3(b)) and the image obtained after normalization(Figure 3(c)) It can be noted that the weighted image showsless intensity variation specifically in those regions with thelowest contribution of the confidence map (yellow andorange areas) As a result intensity variations around thefetal head are attenuated as can be seen by comparing theimages before and after confidence map weighting

To evaluate the performance of RF classification fivelabeled fetal head volumes were used where the task was todiscriminate between voxels corresponding to the fetalcranium (class c +1) and the rest of structures (classc minus 1) For this 5-fold crossvalidation was used Seg-mentation results for the five evaluated volumes are pre-sented in Table 1 It can be seen that all metrics showconsistent results with standard deviations of 35 1 and34 for the Dice HSD and AUC indices respectively Inaddition the average performances obtained by Dice andAUC metrics are higher than 80 which is consideredacceptable to generate the point clouds and the probabilisticmaps needed in the registration process

In Figure 4(a) an example of fetal cranium segmentationis presented where it can be seen that it contains severalmissing regions due to US acoustic shadows As alreadymentioned these missing data hinder the alignment processIn addition Figure 4(b) shows an example of the same pointcloud with the weighting factors obtained using RF It can benoticed that the distribution of weights is not uniform dueto the fact that not all the points have the same probability ofbelonging to the fetal cranium As detailed in the meth-odology section these weights are used as membershipprobabilities of the GMM used in the CPD method

-e visual outcome of each methodrsquos registration isshown in Figure 5 Volumes to be aligned were acquired inan axial and coronal view (Figures 5(a) and 5(b) respec-tively) To better appreciate registration results the ref-erence volume is shown in red (axial view) and the alignedvolume is presented in blue (coronal view) In all casesbecause both registered volumes contain different andcomplementary information differences in the visual re-sults of the registration are shown However a goodalignment result can be seen in structures such as cranialcircumference and ventricles In the second row resultsobtained with intensity-based methods are shown wherethe best performance corresponded to the MI algorithm-ese same methods combined with a binary mask pro-vided the registrations presented in the third row(Figures 5(f ) and 5(h)) It can be noticed that MI and CCimproved when using the binary mask contrary to MSE-e fourth row of Figure 5 shows the results of geometric-based methods where RANSAC presented the best per-formance observed with a better adjustment of the fetalcranial circumference Finally in the last row a compar-ison of the CPD method (Figure 5(l)) with the proposedmethod (PL-CPD) (Figure 5(m)) can be appreciated It canbe noted that performance measured by TRE of PL-CPD isbetter compared to CPD -e corresponding registeredimages show a good alignment of fetal heads and of thecerebral ventriclesrsquo central region

Visual results of the registered point clouds using CPDand PL-CPD methods can be seen in Figure 6 in red thefixed set of points and in blue the aligned points cloud It canbe observed that the PL-CPD method shows a slight im-provement in aligning both studies (Figure 6(b)) comparedto the CPD method (Figure 6(a)) -is can be due to thecontribution of the nonuniform assignment membershipprobabilities of the GMM

To perform a quantitative evaluation of the obtainedresults as well as a comparison with other registrationmethods commonly used in the literature the TRE wascalculated between the reference volume and the study to beregistered-e results are shown in Table 2 To determine thestatistically significant differences between the proposed andthe other methods a paired Student t-test (plt 005) wascarried out for each comparison -e first section of Table 2shows the results in millimeters of several intensity-basedalgorithms where MI presented the smallest error followedby MSE and CC all of them showed statistical differencescompared with the proposed method As expected thesemethods based entirely on intensity information presentconsiderable errors when aligning US fetal brain studies withacoustic shadows and artefacts that alter the registrationresult

As described in the methodology section the intensity-based methods were combined with a binary mask productof fetal head segmentation In this way registration wascarried out only with visible information without consid-ering acoustic shadows -e results are shown in the secondsection of Table 2 A general improvement can be observedin each method obtaining a 78 TRE reduction for MI-BM81 for CC-BM and 92 forMSE-BM compared with their

Computational and Mathematical Methods in Medicine 7

corresponding versions without using a binary mask De-spite these improvements these three methods presentstatistical differences with respect to the proposed methodBinary masks helped to attenuate artefacts introduced byacoustic shadows in the registration process but becauseboth registered studies do not have the same informationthe alignment result is affected

In the last section of Table 2 results obtained with geo-metric-based methods are presented of which ICP(978plusmn 465mm) SIFT (1028plusmn 483mm) and RANSAC(773plusmn 457mm) show statistical differences compared to PL-CPD In contrast CPD presents a small error with689plusmn 408mm Finally the result obtained with the proposedmethod shows the best performance of all with638plusmn 324mm which outperforms the CPD algorithm -isindicates that the incorporation of probabilistic weights asprobability of belonging contributes to the alignment process

Additionally computing times for each of the comparedalgorithms were measured and are shown in the last column

of Table 2 -e first section shows that intensity-basedmethods present the highest times with 375 plusmn 93min forMI 499 plusmn 132min for CC and 396 plusmn 161min for MSEcomputations When binary masks are incorporated to thesealgorithms times are considerably reduced down to 567519 and 566 for MI-BM CC-BM and MSE-BM re-spectively In contrast computation times for geometrymethods are inferior to those obtained by intensity-basedstrategies being 164 plusmn 88min for RANSAC followed bySIFT with 143 plusmn 62min -e PL-CPD algorithm shows anaverage computational time of 127 plusmn 48min which repre-sents an increase of 47 with respect to CPD Finally ICPtook the best computation time with 95 plusmn 27min but it alsopresents considerable registration errors-ese results concurwith a similar evaluation reported by Savva et al [11] where itis concluded that intensity methods are heavier in compu-tational cost because of a wider search space for optimalregistration Another aspect to be considered is that several ofthe compared correspondence-based and intensity-basedmethods combined with binary mask (marked with + inTable 2) require a previous fetal cranium segmentation withRF which adds 74 plusmn 26min to overall computation timesFurthermore given that RF is a supervised classifier it re-quires a previous training step (described in Section 22) thatrepresents an average time of 128 plusmn 34min -is step iscarried out only one and therefore it is not included incomputational times reported in Table 2

In a preliminary study an SVM classifier to segment fetalcranium was proposed -e TRE obtained by SVM was621 plusmn 378mm which is similar to the TRE shown by RF

250

200

150

100

50

0Grayscale

(a)

Confidencefactor

1

08

06

04

02

0

(b)250

200

150

100

50

0Grayscale

(c)

Figure 3 Example of the US imagesrsquo weighting using confidence maps (a) representative fetal head US image in axial view (b) cor-responding confidence map and (c) weighted image

Table 1 Quantitative evaluation of fetal head segmentation usingRF classifier

No Dice () HSD (mm) AUC ()1 913 48 8822 867 57 8243 886 51 8544 854 66 8355 82 73 792Global (μ plusmn σ) 868 plusmn 35 59 plusmn 1 837 plusmn 34

8 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

neighboring points and were incorporated independentlyinto a probabilistic cost function -e results show im-provements in the CPD method however it is necessary tocombine these indices to achieve better performance Zhanget al [21] incorporated SURF as a preliminary step to aligncontours afterwards the membership probability of theGMM is estimated by the edge point confidence -is CPDvariant was applied to synthetic aperture radar imagesobtaining better results compared to the original method Luet al [22] proposed the accelerated CPD for fast registrationof 3D point clouds -ese authors incorporated globalsquared iterative expectation-maximization algorithm withfast Gauss transform to the CPD method managing toreduce the processing time up to ten times Other authorshave reported CPD variants seeking to optimally assign theGMM membership probability using rotation invariantshape descriptors [23] correspondence priors and corre-spondence preserving subsampling approaches [24] and theshape context of one point with respect to the distribution ofother points [25] On the other hand Saval-Calvo et al [26]proposed color-CPD algorithm to register 3D points byusing color and shape spaces to jointly estimate the bestmatch -ese authors consider that the incorporation ofcolor helps to register sets of points with noise or missingdata -e color-CPD combines the color information in theiterative process of aligning two sets of points However thismethod considers an isotropic covariance matrix and anequal membership probability in the GMM In general thesemethods report variations of the CPD algorithm seeking toimprove the computational speed the automatic weightingof the membership probability and the weight factor defi-nition to deal with outliers and missing data for differentapplications of 2D and 3D image registration A disad-vantage to highlight is that these methods need an initialpoint cloud prior to registration which is not available in USfetal brain studies because they are constituted by intensityvoxels

For the specific application of 3D US fetal head align-ment several works have been published -is is a difficulttask due tomultiplicative noise fetal movement and acousticshadows and factors that hinder an adequate alignment Inthis context Cen et al [27 28] have worked on the regis-tration of volumes using shape and texture patternsextracted from a Gabor filter bank -is set of features wasused to register studies in 2D and 3D using only one fetalbrain Fathima et al [29] reported a method based onrepresenting each image by amplitude orientation and localphase It is used to perform the affine registration based onnormalized mutual information Chen et al [30] proposedan algorithm for fetal head registration of US phantomvolumes It is based on previously matching segmentedfeatures such as the eyes or the fetal head using feature-basedregistration

In other image modalities Kuklisova et al [31] reporteda registration method between fetal brain MRI and USstudies in which the former was converted into a pseu-dovolume of US using a probabilistic atlas Subsequently thepseudovolume is aligned towards a conventional US fetalbrain study using a robust block-rigid registration -e aim

of this work was to use the information from the registeredMRI study to improve the visualization of fetal brain ana-tomical structures All these works seek to register fetalbrain studies of US-US and US-MRI in two and three di-mensions However some of them use synthetic images orphantoms Additionally in none of the cases do they con-sider studies with artefacts of fetal brain occlusion whichconsiderably affects the registration methods based on in-tensity or texture

An exhaustive review of US image registration waspublished by [32] -e authors include image-basedmethods such as MI CC or MSE correspondence-basedmethods where characteristics were selected either manu-ally or using algorithms like SIFT or other similaritymeasures such as Hellinger distance statistics-based fuzzylocal binary patterns or the sum of absolute differencesAccording to this review the most important drawbacks forUS registration are multiplicative speckle and occlusionartefacts while the main reported medical imaging appli-cations are in head neck breast heart liver kidney boneprostate and fetal studies

To the best of our knowledge there are not any researchstudies that address the problem of automatic rigid regis-tration of 3D US fetal heads based on point clouds that canwork correctly in the presence of artefacts such as multi-plicative noise occluded brain regions and outliers

In the present work a registration strategy is proposedthat has two main contributions

(1) Automatic registration of real 3D US fetal brainvolumes contains a considerable degree of occlusionartefacts noise and missing data Previous researchstudies have not addressed this issue in a real clinicalcontext As already mentioned the alignment of fetalstudies can be useful in several relevant clinicalapplications such as brain structures segmentationfetal growth monitoring multiprojection fusion oracoustic shadows mitigation

(2) A variation of the CPD method that incorporatesnew features automatic segmentation and confor-mation of the point cloud qualities that have notbeen reported by any previously developed CPDvariant together with an appropriate proposedweighting of membership probabilities of the GMMSegmentation as well as weighting factors are ob-tained from an RF-based classifier fed by a set offeatures composed of intensity texture and edgeparameters Preliminary results of this research havebeen reported in [33] in which a variant of the CPDmethod using another scheme of weighting waspresented

In Section 2 the registration methodology as well as thecharacteristics of the fetal studies used for training andvalidation is described in detail In Section 3 the obtainedresults the corresponding discussion and the comparisonagainst other intensity-based and geometric-based regis-tration methods are exposed Finally in Section 4 theconclusions drawn from this work are presented

Computational and Mathematical Methods in Medicine 3

2 Material and Method

-e overall diagram of the proposed methodology is shownin Figure 1 and will be described in detail in the followingsections It starts with the acquisition of multiple US vol-umes of the fetal head observed at different projections(Figure 1(a) Section 21) followed by a preprocessing stepwhere confidence maps and texture features are computed(Figure 1(b)) -ese parameters are used for segmentationpurposes and to estimate probabilistic weights by means ofRandom Forest classification (Figure 1(c) Section 22) -epoint clouds built from the segmented fetal head are thenweighted with the posterior probability values estimatedfrom the Random Forest process (Figure 1(d) Section 22) tofinally carry out the registration between the two pointclouds using the proposed method (Figure 1(e)) as de-scribed in Section 23

21 Fetal Brain Data Set Eighteen fetal brains between 20and 24 weeks of gestational age were acquired using acurvilinear ultrasound transducer in mode B at frequenciesof 8ndash20MHz (Voluson E8 General Electric HealthcareCompany USA) All volumes had isotropic resolutionsgoing from 02 to 05mm3 Each case consisted on twostudies taken at coronal and axial projections to the samepatient in equal conditions All studies were acquired byobstetrics experts in Fetal Medicine and were approved bythe Department of Fetal Medicine of the National Instituteof Perinatology (INPer) All patients gave their consentaccording to the Declaration of Helsinki Finally each pair ofvolumes was manually aligned by experts in obstetrics fromINPer for this several fiduciary points were considered suchas middle line peduncles and thalamus among others -isstep will be important for the evaluation and comparison ofthe proposed registration method

22RandomForestProbabilisticWeights As a preprocessingstep the confidence map was calculated for each US ac-quisition -e purpose is to emphasize shaded or attenuatedregions in US data obtaining images with homogeneousintensities It has been proven that these maps can help in theprocessing and registration of US data [34]

Subsequently a binary classification scheme based onRandom Forest (RF) which is a machine learning algorithmbased on the ensemble of multiple decision trees [35] wasimplemented to find the weighting factors In a previousresearch a Support Vector Machine (SVM) classifier wastested for this purpose [33] with similar performance butwith a higher computational cost

Given a training data set V (x1 x2 xd) isin Rd be-longing to two classes c +1 c minus 1 where V representseach voxel xd are the responses of a chosen filter bank at thatparticular voxel and d is the feature number or extractedpatterns in each voxel -e task is to optimize an energyfunction through the training set S0 (root node) associatedwith a labeled data set -e optimization function can bedefined as follows

βj argmaxβisinψj

I Sj θ1113872 1113873 (1)

where for each subset of input training data Sj RF learns thefunction that best splits Sj into left SL

j and right SRj nodes In

this context j represents each division node β is a set of splitparameters and ψj is a random subset of the entire pa-rameter space ψ -ese parameters help to reduce possibleoverfitting thus improving the generalization capabilities ofRF For classification purposes the function I can be definedas the gain information for discrete distributions

I Sj β1113872 1113873 H Sj1113872 1113873 minus 1113944iisin LR

Sij

11138681113868111386811138681113868

11138681113868111386811138681113868

Sj

11138681113868111386811138681113868

11138681113868111386811138681113868H S

ij1113872 1113873 (2)

with i indexing the two child nodes (left L or right R node)and H(Si

j) being the entropy of a given subset Sij which can

be defined as follows

H Sij1113872 1113873 minus 1113944

cisinplusmn1P(c)logP(c) (3)

where p(c) is calculated as the normalized empirical his-togram of labels corresponding to the training points in Si

jTo assemble the result of all the decision trees given the classp(c +1 |V) the content result of each leaf is accumulatedby

P(c +1 |V) 1T

1113944

T

t1Pt(c +1 |V) (4)

where t 1 2 T represents the number of trees used inthe classification and P(c +1 |V) is denoted as the posteriorprobability of the classification process For this work the fetalcranium is considered as class c +1 (segmentation requiredto build the point cloud) and the rest of brain tissue andacoustic artefacts as class as it can be seen in Figure 2

-e task is to automatically classify these two classesusing a set of features that includes intensity informationand several texture filters such as variance rank entropymedian and Wiener estimated on an (9 times 9 times 9) analysiswindow [36] Other incorporated features were the edgesobtained from Canny [37] and Laplacian filtering Followingthe fetal cranium segmentation the point cloud is con-structed using constrained Delaunay tetrahedralization [38]-e point clouds are the vertices in cartesian coordinates ofthe obtained triangular mesh and correspond to the innerand outer enveloping areas of the fetal cranium

23 Probabilistic Learning Coherent Point Drift -e methodproposed for fetal brain volume registration is based on thealgorithm developed byMyronenko and Song [16] known asCoherent Point Drift It consists of a geometric registrationthat establishes point to point correspondences and assumesthat each element drifts coherently as a group while pre-serving the pointsrsquo cluster topology Let X and Y be two setsof points to be registered -ey can be denoted asY ym | 1 2 M1113864 1113865 and X xn | 1 2 N1113864 1113865 where Ycorresponds to the set of centroids of the GMM and X is the

4 Computational and Mathematical Methods in Medicine

generated pointsrsquo set -e task is to obtain an affine trans-form T such that X T(YΘ) where Θ is a matrixcontaining rotation translation and scaling parameters-eprobability density function (PDF) on a GMM can bewritten as follows

p(x) 1113944M+1

m1P(m)p(x | m) (5)

where

p(x | m) 1

2πσ2exp minus

x minus T ym θ( 11138571113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138682

2σ2⎧⎨

⎩

⎫⎬

⎭ m 1 2 M

(6)

and P(m) (1M) is an equiprobable distributionA uniform distribution denoted by p(x | M + 1) 1N

is used to deal with outliers occlusion artefacts and noise Aweight factor (0ltwlt 1) representing the outliersrsquo ratio isincorporated into the distribution together with an isotropicvariance σ2 term -erefore the PDF is formulated asfollows

p(x) ω1N

+(1 minus ω) 1113944M+1

m1

1M

p(x | m) (7)

Parametersrsquo matrix Θ is derived from the GMM cen-troids and can be estimated by probability maximization orby minimization of the negative logarithmic probability ofthe following function

E Θ σ21113872 1113873 minus 1113944

N

n1log 1113944

M+1

m1P(m)p xn

1113868111386811138681113868 m1113872 1113873 (8)

previously assuming that each gaussian component isindependent

Given two points sets (ym and xn) the GMM posteriorprobability is denoted by p(m | xn) P(m)p(xn | m)p(xn)-e CPD method applies an Expectation-Maximization(EM) algorithm to adjust Θ and σ2 using Bayesrsquo theorem-e posterior probabilities of the mixture componentsPold(m | xn) (equation (9)) are iteratively computed in whatis known as the expectation or E-step

Pold

m | xn( 1113857 exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967

1113936Mk1exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M (9)

tree T

tree T

3D US fetalhead acquisitions

Preprocessing andtexture

maps analysis

Fetal head segmentationand weight estimation

by random forest

Weighted pointclouds of the

fetal head

Fetal headsaligned by PL-CPD

(a) (b) (c) (d) (e)

Figure 1 Methodological diagram

250

200

150

100

50

0c = +1

c = ndash1Grayscale

Figure 2 Classes considered for fetal cranium segmentation

Computational and Mathematical Methods in Medicine 5

-e ldquonewrdquo parameter values are estimated byminimizingexpectation of the complete negative log-likelihoodfunction

Q minus 1113944N

n11113944

M+1

m1Pold

m | xn( 1113857log Pnew

(m)pnew

xn

1113868111386811138681113868 m1113872 11138731113960 1113961

(10)

during the maximization or M-step -e new Θ and σ2parameters are obtained by rewriting equation (10)

Q Θ σ21113872 1113873 12σ2

1113944

N

n11113944

M+1

m1Pold

m | xn( 1113857 xn minus T ym θ( 1113857

2

+DNp

2log σ2

(11)

where D corresponds to data dimensionality -e algorithmalternatively iterates E-step and M-step until Qrsquos conver-gence thus aligning the sets of points with X T(YΘ)

However assigning the same probability P(m) (1M)

to all GMM components can affect registration quality given

that each Gaussian contributes differently in the model Inthis work it is proposed to weight the membership prob-abilities with the values obtained from the posterior prob-ability resulting from the RF classification process At thesame time the fetal cranium classification helps to obtain thepoint clouds used in the registration

In this way the GMM membership probabilities P(m)

correspond to the posterior probability (normalized between0 and 1) obtained during the classification stepP(c +1 | xn) (equation (4)) (where c +1 is the classcorresponding to fetal cranium) for each point xn It can beexpressed by

P(m)lowast

P c +1 | xn( 1113857

λ (12)

where

λ 1113944M

k1P c +1 | xn( 1113857k (13)

-erefore Pold(m | xn) and the Q objective function canbe rewritten as follows

Poldlowast

m | xn( 1113857 P c +1 | xn( 1113857exp minus xn minus T ym θ( 1113857

22σ21113874 11138751113882 1113883

1113936Mk1P c +1 | xn( 1113857k exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M

Qlowast

Q + 1113944N

n11113944

M

m1Pold

m | xn( 1113857logP(m)lowast

(14)

With this weighting strategy outliers will have less contributioncompared to voxels that describe the fetal cranium which willhave a larger weight during the registration process -esechanges do not affect the previously described EM optimi-zation procedure -us the Probabilistic Learning CoherentPoint Drift (PL-CPD) method has the advantages of being lesssensitive to outliers and do not requiring a high computationalcost compared to other intensity-based methods

24 Evaluation of the PL-CPD Method To assess RFrsquos per-formance in head segmentation a 5-fold crossvalidation wascarried out where the training and test data sets were ob-tained from a sample of five US volumes (an example of theclasses is presented in Figure 2) As segmentation perfor-mance metric the Dice index was computed which is de-fined as follows

Dice 2|AcapB|

|A| +|B| (15)

where A is the volume segmented by an expert in obstetricsand B represents the automatically segmented volumeAdditionally the Hausdorff Surface Distance (HSD) [39]and the area under the receiver operating characteristic

(ROC) curve (AUC) were obtained After segmentationvalidation the RF was applied to the remaining studies

Subsequently to validate the proposed PL-CPDmethodthe Target Registration Error (TRE) (equation (16)) betweenthe reference volume VR and the study to be aligned VM

were measured considering the eighteen pairs of registeredvolumes A total of n 1000 uniformly distributed targetpoints were defined as ai and bi corresponding to theregistered and reference volumes respectively

TRE VR VM( 1113857

1n

1113944

n

i1bi minus ai( 1113857

2

11139741113972

(16)

According to Fitzpatrick [40] TRE provides a betterestimation of registration errors than a landmark-basedmetric Additionally to evaluate in detail translation androtation errors the Root Mean Square (RMS) difference wasused It can be denoted as follows

RMS VR VM( 1113857

1m

1113944

m

k1bk minus ak( 1113857

2

11139741113972

(17)

To measure displacement errors the values in the threeaxes (ΔX ΔY and ΔZ) are considered as bk and ak with

6 Computational and Mathematical Methods in Medicine

m 3 on the other hand for the rotation errors the values ofthe angles (α δ and λ) are taken as bk and ak

To evaluate its performance the proposed PL-CPD al-gorithm was compared with other previously reported in-tensity-based registration methods Mutual Information(MI) Cross Correlation (CC) and Mean Squared Error(MSE) [6ndash8] -e method was also compared with somegeometric registration methods based on correspondencesuch as Iterative Closest Point (ICP) [12 13] RandomSample Consensus (RANSAC) [14] Scale-Invariant FeatureTransform (SIFT) [15] and Coherent Point Drift (CPD)[16] For all comparisons TRErsquos mean and standard devi-ation were obtained from the accumulated errors measuredfor each registration case Statistical differences for thesetests were determined by an paired Student t-test (plt 005)comparing the PL-CPD method with each other In addi-tion the mean RMS errors for translations and rotationswere obtained for each carried out registration

3 Results and Discussion

In this section results of the proposed registration method(PL-CPD) as well as the corresponding discussion arepresented As mentioned in Section 22 the procedure be-gins with a preprocessing step of background normalizationwith confidence maps An example of the results obtained isshown in Figure 3 fetal head image in axial view(Figure 3(a)) its corresponding confidence map(Figure 3(b)) and the image obtained after normalization(Figure 3(c)) It can be noted that the weighted image showsless intensity variation specifically in those regions with thelowest contribution of the confidence map (yellow andorange areas) As a result intensity variations around thefetal head are attenuated as can be seen by comparing theimages before and after confidence map weighting

To evaluate the performance of RF classification fivelabeled fetal head volumes were used where the task was todiscriminate between voxels corresponding to the fetalcranium (class c +1) and the rest of structures (classc minus 1) For this 5-fold crossvalidation was used Seg-mentation results for the five evaluated volumes are pre-sented in Table 1 It can be seen that all metrics showconsistent results with standard deviations of 35 1 and34 for the Dice HSD and AUC indices respectively Inaddition the average performances obtained by Dice andAUC metrics are higher than 80 which is consideredacceptable to generate the point clouds and the probabilisticmaps needed in the registration process

In Figure 4(a) an example of fetal cranium segmentationis presented where it can be seen that it contains severalmissing regions due to US acoustic shadows As alreadymentioned these missing data hinder the alignment processIn addition Figure 4(b) shows an example of the same pointcloud with the weighting factors obtained using RF It can benoticed that the distribution of weights is not uniform dueto the fact that not all the points have the same probability ofbelonging to the fetal cranium As detailed in the meth-odology section these weights are used as membershipprobabilities of the GMM used in the CPD method

-e visual outcome of each methodrsquos registration isshown in Figure 5 Volumes to be aligned were acquired inan axial and coronal view (Figures 5(a) and 5(b) respec-tively) To better appreciate registration results the ref-erence volume is shown in red (axial view) and the alignedvolume is presented in blue (coronal view) In all casesbecause both registered volumes contain different andcomplementary information differences in the visual re-sults of the registration are shown However a goodalignment result can be seen in structures such as cranialcircumference and ventricles In the second row resultsobtained with intensity-based methods are shown wherethe best performance corresponded to the MI algorithm-ese same methods combined with a binary mask pro-vided the registrations presented in the third row(Figures 5(f ) and 5(h)) It can be noticed that MI and CCimproved when using the binary mask contrary to MSE-e fourth row of Figure 5 shows the results of geometric-based methods where RANSAC presented the best per-formance observed with a better adjustment of the fetalcranial circumference Finally in the last row a compar-ison of the CPD method (Figure 5(l)) with the proposedmethod (PL-CPD) (Figure 5(m)) can be appreciated It canbe noted that performance measured by TRE of PL-CPD isbetter compared to CPD -e corresponding registeredimages show a good alignment of fetal heads and of thecerebral ventriclesrsquo central region

Visual results of the registered point clouds using CPDand PL-CPD methods can be seen in Figure 6 in red thefixed set of points and in blue the aligned points cloud It canbe observed that the PL-CPD method shows a slight im-provement in aligning both studies (Figure 6(b)) comparedto the CPD method (Figure 6(a)) -is can be due to thecontribution of the nonuniform assignment membershipprobabilities of the GMM

To perform a quantitative evaluation of the obtainedresults as well as a comparison with other registrationmethods commonly used in the literature the TRE wascalculated between the reference volume and the study to beregistered-e results are shown in Table 2 To determine thestatistically significant differences between the proposed andthe other methods a paired Student t-test (plt 005) wascarried out for each comparison -e first section of Table 2shows the results in millimeters of several intensity-basedalgorithms where MI presented the smallest error followedby MSE and CC all of them showed statistical differencescompared with the proposed method As expected thesemethods based entirely on intensity information presentconsiderable errors when aligning US fetal brain studies withacoustic shadows and artefacts that alter the registrationresult

As described in the methodology section the intensity-based methods were combined with a binary mask productof fetal head segmentation In this way registration wascarried out only with visible information without consid-ering acoustic shadows -e results are shown in the secondsection of Table 2 A general improvement can be observedin each method obtaining a 78 TRE reduction for MI-BM81 for CC-BM and 92 forMSE-BM compared with their

Computational and Mathematical Methods in Medicine 7

corresponding versions without using a binary mask De-spite these improvements these three methods presentstatistical differences with respect to the proposed methodBinary masks helped to attenuate artefacts introduced byacoustic shadows in the registration process but becauseboth registered studies do not have the same informationthe alignment result is affected

In the last section of Table 2 results obtained with geo-metric-based methods are presented of which ICP(978plusmn 465mm) SIFT (1028plusmn 483mm) and RANSAC(773plusmn 457mm) show statistical differences compared to PL-CPD In contrast CPD presents a small error with689plusmn 408mm Finally the result obtained with the proposedmethod shows the best performance of all with638plusmn 324mm which outperforms the CPD algorithm -isindicates that the incorporation of probabilistic weights asprobability of belonging contributes to the alignment process

Additionally computing times for each of the comparedalgorithms were measured and are shown in the last column

of Table 2 -e first section shows that intensity-basedmethods present the highest times with 375 plusmn 93min forMI 499 plusmn 132min for CC and 396 plusmn 161min for MSEcomputations When binary masks are incorporated to thesealgorithms times are considerably reduced down to 567519 and 566 for MI-BM CC-BM and MSE-BM re-spectively In contrast computation times for geometrymethods are inferior to those obtained by intensity-basedstrategies being 164 plusmn 88min for RANSAC followed bySIFT with 143 plusmn 62min -e PL-CPD algorithm shows anaverage computational time of 127 plusmn 48min which repre-sents an increase of 47 with respect to CPD Finally ICPtook the best computation time with 95 plusmn 27min but it alsopresents considerable registration errors-ese results concurwith a similar evaluation reported by Savva et al [11] where itis concluded that intensity methods are heavier in compu-tational cost because of a wider search space for optimalregistration Another aspect to be considered is that several ofthe compared correspondence-based and intensity-basedmethods combined with binary mask (marked with + inTable 2) require a previous fetal cranium segmentation withRF which adds 74 plusmn 26min to overall computation timesFurthermore given that RF is a supervised classifier it re-quires a previous training step (described in Section 22) thatrepresents an average time of 128 plusmn 34min -is step iscarried out only one and therefore it is not included incomputational times reported in Table 2

In a preliminary study an SVM classifier to segment fetalcranium was proposed -e TRE obtained by SVM was621 plusmn 378mm which is similar to the TRE shown by RF

250

200

150

100

50

0Grayscale

(a)

Confidencefactor

1

08

06

04

02

0

(b)250

200

150

100

50

0Grayscale

(c)

Figure 3 Example of the US imagesrsquo weighting using confidence maps (a) representative fetal head US image in axial view (b) cor-responding confidence map and (c) weighted image

Table 1 Quantitative evaluation of fetal head segmentation usingRF classifier

No Dice () HSD (mm) AUC ()1 913 48 8822 867 57 8243 886 51 8544 854 66 8355 82 73 792Global (μ plusmn σ) 868 plusmn 35 59 plusmn 1 837 plusmn 34

8 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

2 Material and Method

-e overall diagram of the proposed methodology is shownin Figure 1 and will be described in detail in the followingsections It starts with the acquisition of multiple US vol-umes of the fetal head observed at different projections(Figure 1(a) Section 21) followed by a preprocessing stepwhere confidence maps and texture features are computed(Figure 1(b)) -ese parameters are used for segmentationpurposes and to estimate probabilistic weights by means ofRandom Forest classification (Figure 1(c) Section 22) -epoint clouds built from the segmented fetal head are thenweighted with the posterior probability values estimatedfrom the Random Forest process (Figure 1(d) Section 22) tofinally carry out the registration between the two pointclouds using the proposed method (Figure 1(e)) as de-scribed in Section 23

21 Fetal Brain Data Set Eighteen fetal brains between 20and 24 weeks of gestational age were acquired using acurvilinear ultrasound transducer in mode B at frequenciesof 8ndash20MHz (Voluson E8 General Electric HealthcareCompany USA) All volumes had isotropic resolutionsgoing from 02 to 05mm3 Each case consisted on twostudies taken at coronal and axial projections to the samepatient in equal conditions All studies were acquired byobstetrics experts in Fetal Medicine and were approved bythe Department of Fetal Medicine of the National Instituteof Perinatology (INPer) All patients gave their consentaccording to the Declaration of Helsinki Finally each pair ofvolumes was manually aligned by experts in obstetrics fromINPer for this several fiduciary points were considered suchas middle line peduncles and thalamus among others -isstep will be important for the evaluation and comparison ofthe proposed registration method

22RandomForestProbabilisticWeights As a preprocessingstep the confidence map was calculated for each US ac-quisition -e purpose is to emphasize shaded or attenuatedregions in US data obtaining images with homogeneousintensities It has been proven that these maps can help in theprocessing and registration of US data [34]

Subsequently a binary classification scheme based onRandom Forest (RF) which is a machine learning algorithmbased on the ensemble of multiple decision trees [35] wasimplemented to find the weighting factors In a previousresearch a Support Vector Machine (SVM) classifier wastested for this purpose [33] with similar performance butwith a higher computational cost

Given a training data set V (x1 x2 xd) isin Rd be-longing to two classes c +1 c minus 1 where V representseach voxel xd are the responses of a chosen filter bank at thatparticular voxel and d is the feature number or extractedpatterns in each voxel -e task is to optimize an energyfunction through the training set S0 (root node) associatedwith a labeled data set -e optimization function can bedefined as follows

βj argmaxβisinψj

I Sj θ1113872 1113873 (1)

where for each subset of input training data Sj RF learns thefunction that best splits Sj into left SL

j and right SRj nodes In

this context j represents each division node β is a set of splitparameters and ψj is a random subset of the entire pa-rameter space ψ -ese parameters help to reduce possibleoverfitting thus improving the generalization capabilities ofRF For classification purposes the function I can be definedas the gain information for discrete distributions

I Sj β1113872 1113873 H Sj1113872 1113873 minus 1113944iisin LR

Sij

11138681113868111386811138681113868

11138681113868111386811138681113868

Sj

11138681113868111386811138681113868

11138681113868111386811138681113868H S

ij1113872 1113873 (2)

with i indexing the two child nodes (left L or right R node)and H(Si

j) being the entropy of a given subset Sij which can

be defined as follows

H Sij1113872 1113873 minus 1113944

cisinplusmn1P(c)logP(c) (3)

where p(c) is calculated as the normalized empirical his-togram of labels corresponding to the training points in Si

jTo assemble the result of all the decision trees given the classp(c +1 |V) the content result of each leaf is accumulatedby

P(c +1 |V) 1T

1113944

T

t1Pt(c +1 |V) (4)

where t 1 2 T represents the number of trees used inthe classification and P(c +1 |V) is denoted as the posteriorprobability of the classification process For this work the fetalcranium is considered as class c +1 (segmentation requiredto build the point cloud) and the rest of brain tissue andacoustic artefacts as class as it can be seen in Figure 2

-e task is to automatically classify these two classesusing a set of features that includes intensity informationand several texture filters such as variance rank entropymedian and Wiener estimated on an (9 times 9 times 9) analysiswindow [36] Other incorporated features were the edgesobtained from Canny [37] and Laplacian filtering Followingthe fetal cranium segmentation the point cloud is con-structed using constrained Delaunay tetrahedralization [38]-e point clouds are the vertices in cartesian coordinates ofthe obtained triangular mesh and correspond to the innerand outer enveloping areas of the fetal cranium

23 Probabilistic Learning Coherent Point Drift -e methodproposed for fetal brain volume registration is based on thealgorithm developed byMyronenko and Song [16] known asCoherent Point Drift It consists of a geometric registrationthat establishes point to point correspondences and assumesthat each element drifts coherently as a group while pre-serving the pointsrsquo cluster topology Let X and Y be two setsof points to be registered -ey can be denoted asY ym | 1 2 M1113864 1113865 and X xn | 1 2 N1113864 1113865 where Ycorresponds to the set of centroids of the GMM and X is the

4 Computational and Mathematical Methods in Medicine

generated pointsrsquo set -e task is to obtain an affine trans-form T such that X T(YΘ) where Θ is a matrixcontaining rotation translation and scaling parameters-eprobability density function (PDF) on a GMM can bewritten as follows

p(x) 1113944M+1

m1P(m)p(x | m) (5)

where

p(x | m) 1

2πσ2exp minus

x minus T ym θ( 11138571113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138682

2σ2⎧⎨

⎩

⎫⎬

⎭ m 1 2 M

(6)

and P(m) (1M) is an equiprobable distributionA uniform distribution denoted by p(x | M + 1) 1N

is used to deal with outliers occlusion artefacts and noise Aweight factor (0ltwlt 1) representing the outliersrsquo ratio isincorporated into the distribution together with an isotropicvariance σ2 term -erefore the PDF is formulated asfollows

p(x) ω1N

+(1 minus ω) 1113944M+1

m1

1M

p(x | m) (7)

Parametersrsquo matrix Θ is derived from the GMM cen-troids and can be estimated by probability maximization orby minimization of the negative logarithmic probability ofthe following function

E Θ σ21113872 1113873 minus 1113944

N

n1log 1113944

M+1

m1P(m)p xn

1113868111386811138681113868 m1113872 1113873 (8)

previously assuming that each gaussian component isindependent

Given two points sets (ym and xn) the GMM posteriorprobability is denoted by p(m | xn) P(m)p(xn | m)p(xn)-e CPD method applies an Expectation-Maximization(EM) algorithm to adjust Θ and σ2 using Bayesrsquo theorem-e posterior probabilities of the mixture componentsPold(m | xn) (equation (9)) are iteratively computed in whatis known as the expectation or E-step

Pold

m | xn( 1113857 exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967

1113936Mk1exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M (9)

tree T

tree T

3D US fetalhead acquisitions

Preprocessing andtexture

maps analysis

Fetal head segmentationand weight estimation

by random forest

Weighted pointclouds of the

fetal head

Fetal headsaligned by PL-CPD

(a) (b) (c) (d) (e)

Figure 1 Methodological diagram

250

200

150

100

50

0c = +1

c = ndash1Grayscale

Figure 2 Classes considered for fetal cranium segmentation

Computational and Mathematical Methods in Medicine 5

-e ldquonewrdquo parameter values are estimated byminimizingexpectation of the complete negative log-likelihoodfunction

Q minus 1113944N

n11113944

M+1

m1Pold

m | xn( 1113857log Pnew

(m)pnew

xn

1113868111386811138681113868 m1113872 11138731113960 1113961

(10)

during the maximization or M-step -e new Θ and σ2parameters are obtained by rewriting equation (10)

Q Θ σ21113872 1113873 12σ2

1113944

N

n11113944

M+1

m1Pold

m | xn( 1113857 xn minus T ym θ( 1113857

2

+DNp

2log σ2

(11)

where D corresponds to data dimensionality -e algorithmalternatively iterates E-step and M-step until Qrsquos conver-gence thus aligning the sets of points with X T(YΘ)

However assigning the same probability P(m) (1M)

to all GMM components can affect registration quality given

that each Gaussian contributes differently in the model Inthis work it is proposed to weight the membership prob-abilities with the values obtained from the posterior prob-ability resulting from the RF classification process At thesame time the fetal cranium classification helps to obtain thepoint clouds used in the registration

In this way the GMM membership probabilities P(m)

correspond to the posterior probability (normalized between0 and 1) obtained during the classification stepP(c +1 | xn) (equation (4)) (where c +1 is the classcorresponding to fetal cranium) for each point xn It can beexpressed by

P(m)lowast

P c +1 | xn( 1113857

λ (12)

where

λ 1113944M

k1P c +1 | xn( 1113857k (13)

-erefore Pold(m | xn) and the Q objective function canbe rewritten as follows

Poldlowast

m | xn( 1113857 P c +1 | xn( 1113857exp minus xn minus T ym θ( 1113857

22σ21113874 11138751113882 1113883

1113936Mk1P c +1 | xn( 1113857k exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M

Qlowast

Q + 1113944N

n11113944

M

m1Pold

m | xn( 1113857logP(m)lowast

(14)

With this weighting strategy outliers will have less contributioncompared to voxels that describe the fetal cranium which willhave a larger weight during the registration process -esechanges do not affect the previously described EM optimi-zation procedure -us the Probabilistic Learning CoherentPoint Drift (PL-CPD) method has the advantages of being lesssensitive to outliers and do not requiring a high computationalcost compared to other intensity-based methods

24 Evaluation of the PL-CPD Method To assess RFrsquos per-formance in head segmentation a 5-fold crossvalidation wascarried out where the training and test data sets were ob-tained from a sample of five US volumes (an example of theclasses is presented in Figure 2) As segmentation perfor-mance metric the Dice index was computed which is de-fined as follows

Dice 2|AcapB|

|A| +|B| (15)

where A is the volume segmented by an expert in obstetricsand B represents the automatically segmented volumeAdditionally the Hausdorff Surface Distance (HSD) [39]and the area under the receiver operating characteristic

(ROC) curve (AUC) were obtained After segmentationvalidation the RF was applied to the remaining studies

Subsequently to validate the proposed PL-CPDmethodthe Target Registration Error (TRE) (equation (16)) betweenthe reference volume VR and the study to be aligned VM

were measured considering the eighteen pairs of registeredvolumes A total of n 1000 uniformly distributed targetpoints were defined as ai and bi corresponding to theregistered and reference volumes respectively

TRE VR VM( 1113857

1n

1113944

n

i1bi minus ai( 1113857

2

11139741113972

(16)

According to Fitzpatrick [40] TRE provides a betterestimation of registration errors than a landmark-basedmetric Additionally to evaluate in detail translation androtation errors the Root Mean Square (RMS) difference wasused It can be denoted as follows

RMS VR VM( 1113857

1m

1113944

m

k1bk minus ak( 1113857

2

11139741113972

(17)

To measure displacement errors the values in the threeaxes (ΔX ΔY and ΔZ) are considered as bk and ak with

6 Computational and Mathematical Methods in Medicine

m 3 on the other hand for the rotation errors the values ofthe angles (α δ and λ) are taken as bk and ak

To evaluate its performance the proposed PL-CPD al-gorithm was compared with other previously reported in-tensity-based registration methods Mutual Information(MI) Cross Correlation (CC) and Mean Squared Error(MSE) [6ndash8] -e method was also compared with somegeometric registration methods based on correspondencesuch as Iterative Closest Point (ICP) [12 13] RandomSample Consensus (RANSAC) [14] Scale-Invariant FeatureTransform (SIFT) [15] and Coherent Point Drift (CPD)[16] For all comparisons TRErsquos mean and standard devi-ation were obtained from the accumulated errors measuredfor each registration case Statistical differences for thesetests were determined by an paired Student t-test (plt 005)comparing the PL-CPD method with each other In addi-tion the mean RMS errors for translations and rotationswere obtained for each carried out registration

3 Results and Discussion

In this section results of the proposed registration method(PL-CPD) as well as the corresponding discussion arepresented As mentioned in Section 22 the procedure be-gins with a preprocessing step of background normalizationwith confidence maps An example of the results obtained isshown in Figure 3 fetal head image in axial view(Figure 3(a)) its corresponding confidence map(Figure 3(b)) and the image obtained after normalization(Figure 3(c)) It can be noted that the weighted image showsless intensity variation specifically in those regions with thelowest contribution of the confidence map (yellow andorange areas) As a result intensity variations around thefetal head are attenuated as can be seen by comparing theimages before and after confidence map weighting

To evaluate the performance of RF classification fivelabeled fetal head volumes were used where the task was todiscriminate between voxels corresponding to the fetalcranium (class c +1) and the rest of structures (classc minus 1) For this 5-fold crossvalidation was used Seg-mentation results for the five evaluated volumes are pre-sented in Table 1 It can be seen that all metrics showconsistent results with standard deviations of 35 1 and34 for the Dice HSD and AUC indices respectively Inaddition the average performances obtained by Dice andAUC metrics are higher than 80 which is consideredacceptable to generate the point clouds and the probabilisticmaps needed in the registration process

In Figure 4(a) an example of fetal cranium segmentationis presented where it can be seen that it contains severalmissing regions due to US acoustic shadows As alreadymentioned these missing data hinder the alignment processIn addition Figure 4(b) shows an example of the same pointcloud with the weighting factors obtained using RF It can benoticed that the distribution of weights is not uniform dueto the fact that not all the points have the same probability ofbelonging to the fetal cranium As detailed in the meth-odology section these weights are used as membershipprobabilities of the GMM used in the CPD method

-e visual outcome of each methodrsquos registration isshown in Figure 5 Volumes to be aligned were acquired inan axial and coronal view (Figures 5(a) and 5(b) respec-tively) To better appreciate registration results the ref-erence volume is shown in red (axial view) and the alignedvolume is presented in blue (coronal view) In all casesbecause both registered volumes contain different andcomplementary information differences in the visual re-sults of the registration are shown However a goodalignment result can be seen in structures such as cranialcircumference and ventricles In the second row resultsobtained with intensity-based methods are shown wherethe best performance corresponded to the MI algorithm-ese same methods combined with a binary mask pro-vided the registrations presented in the third row(Figures 5(f ) and 5(h)) It can be noticed that MI and CCimproved when using the binary mask contrary to MSE-e fourth row of Figure 5 shows the results of geometric-based methods where RANSAC presented the best per-formance observed with a better adjustment of the fetalcranial circumference Finally in the last row a compar-ison of the CPD method (Figure 5(l)) with the proposedmethod (PL-CPD) (Figure 5(m)) can be appreciated It canbe noted that performance measured by TRE of PL-CPD isbetter compared to CPD -e corresponding registeredimages show a good alignment of fetal heads and of thecerebral ventriclesrsquo central region

Visual results of the registered point clouds using CPDand PL-CPD methods can be seen in Figure 6 in red thefixed set of points and in blue the aligned points cloud It canbe observed that the PL-CPD method shows a slight im-provement in aligning both studies (Figure 6(b)) comparedto the CPD method (Figure 6(a)) -is can be due to thecontribution of the nonuniform assignment membershipprobabilities of the GMM

To perform a quantitative evaluation of the obtainedresults as well as a comparison with other registrationmethods commonly used in the literature the TRE wascalculated between the reference volume and the study to beregistered-e results are shown in Table 2 To determine thestatistically significant differences between the proposed andthe other methods a paired Student t-test (plt 005) wascarried out for each comparison -e first section of Table 2shows the results in millimeters of several intensity-basedalgorithms where MI presented the smallest error followedby MSE and CC all of them showed statistical differencescompared with the proposed method As expected thesemethods based entirely on intensity information presentconsiderable errors when aligning US fetal brain studies withacoustic shadows and artefacts that alter the registrationresult

As described in the methodology section the intensity-based methods were combined with a binary mask productof fetal head segmentation In this way registration wascarried out only with visible information without consid-ering acoustic shadows -e results are shown in the secondsection of Table 2 A general improvement can be observedin each method obtaining a 78 TRE reduction for MI-BM81 for CC-BM and 92 forMSE-BM compared with their

Computational and Mathematical Methods in Medicine 7

corresponding versions without using a binary mask De-spite these improvements these three methods presentstatistical differences with respect to the proposed methodBinary masks helped to attenuate artefacts introduced byacoustic shadows in the registration process but becauseboth registered studies do not have the same informationthe alignment result is affected

In the last section of Table 2 results obtained with geo-metric-based methods are presented of which ICP(978plusmn 465mm) SIFT (1028plusmn 483mm) and RANSAC(773plusmn 457mm) show statistical differences compared to PL-CPD In contrast CPD presents a small error with689plusmn 408mm Finally the result obtained with the proposedmethod shows the best performance of all with638plusmn 324mm which outperforms the CPD algorithm -isindicates that the incorporation of probabilistic weights asprobability of belonging contributes to the alignment process

Additionally computing times for each of the comparedalgorithms were measured and are shown in the last column

of Table 2 -e first section shows that intensity-basedmethods present the highest times with 375 plusmn 93min forMI 499 plusmn 132min for CC and 396 plusmn 161min for MSEcomputations When binary masks are incorporated to thesealgorithms times are considerably reduced down to 567519 and 566 for MI-BM CC-BM and MSE-BM re-spectively In contrast computation times for geometrymethods are inferior to those obtained by intensity-basedstrategies being 164 plusmn 88min for RANSAC followed bySIFT with 143 plusmn 62min -e PL-CPD algorithm shows anaverage computational time of 127 plusmn 48min which repre-sents an increase of 47 with respect to CPD Finally ICPtook the best computation time with 95 plusmn 27min but it alsopresents considerable registration errors-ese results concurwith a similar evaluation reported by Savva et al [11] where itis concluded that intensity methods are heavier in compu-tational cost because of a wider search space for optimalregistration Another aspect to be considered is that several ofthe compared correspondence-based and intensity-basedmethods combined with binary mask (marked with + inTable 2) require a previous fetal cranium segmentation withRF which adds 74 plusmn 26min to overall computation timesFurthermore given that RF is a supervised classifier it re-quires a previous training step (described in Section 22) thatrepresents an average time of 128 plusmn 34min -is step iscarried out only one and therefore it is not included incomputational times reported in Table 2

In a preliminary study an SVM classifier to segment fetalcranium was proposed -e TRE obtained by SVM was621 plusmn 378mm which is similar to the TRE shown by RF

250

200

150

100

50

0Grayscale

(a)

Confidencefactor

1

08

06

04

02

0

(b)250

200

150

100

50

0Grayscale

(c)

Figure 3 Example of the US imagesrsquo weighting using confidence maps (a) representative fetal head US image in axial view (b) cor-responding confidence map and (c) weighted image

Table 1 Quantitative evaluation of fetal head segmentation usingRF classifier

No Dice () HSD (mm) AUC ()1 913 48 8822 867 57 8243 886 51 8544 854 66 8355 82 73 792Global (μ plusmn σ) 868 plusmn 35 59 plusmn 1 837 plusmn 34

8 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

generated pointsrsquo set -e task is to obtain an affine trans-form T such that X T(YΘ) where Θ is a matrixcontaining rotation translation and scaling parameters-eprobability density function (PDF) on a GMM can bewritten as follows

p(x) 1113944M+1

m1P(m)p(x | m) (5)

where

p(x | m) 1

2πσ2exp minus

x minus T ym θ( 11138571113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138682

2σ2⎧⎨

⎩

⎫⎬

⎭ m 1 2 M

(6)

and P(m) (1M) is an equiprobable distributionA uniform distribution denoted by p(x | M + 1) 1N

is used to deal with outliers occlusion artefacts and noise Aweight factor (0ltwlt 1) representing the outliersrsquo ratio isincorporated into the distribution together with an isotropicvariance σ2 term -erefore the PDF is formulated asfollows

p(x) ω1N

+(1 minus ω) 1113944M+1

m1

1M

p(x | m) (7)

Parametersrsquo matrix Θ is derived from the GMM cen-troids and can be estimated by probability maximization orby minimization of the negative logarithmic probability ofthe following function

E Θ σ21113872 1113873 minus 1113944

N

n1log 1113944

M+1

m1P(m)p xn

1113868111386811138681113868 m1113872 1113873 (8)

previously assuming that each gaussian component isindependent

Given two points sets (ym and xn) the GMM posteriorprobability is denoted by p(m | xn) P(m)p(xn | m)p(xn)-e CPD method applies an Expectation-Maximization(EM) algorithm to adjust Θ and σ2 using Bayesrsquo theorem-e posterior probabilities of the mixture componentsPold(m | xn) (equation (9)) are iteratively computed in whatis known as the expectation or E-step

Pold

m | xn( 1113857 exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967

1113936Mk1exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M (9)

tree T

tree T

3D US fetalhead acquisitions

Preprocessing andtexture

maps analysis

Fetal head segmentationand weight estimation

by random forest

Weighted pointclouds of the

fetal head

Fetal headsaligned by PL-CPD

(a) (b) (c) (d) (e)

Figure 1 Methodological diagram

250

200

150

100

50

0c = +1

c = ndash1Grayscale

Figure 2 Classes considered for fetal cranium segmentation

Computational and Mathematical Methods in Medicine 5

-e ldquonewrdquo parameter values are estimated byminimizingexpectation of the complete negative log-likelihoodfunction

Q minus 1113944N

n11113944

M+1

m1Pold

m | xn( 1113857log Pnew

(m)pnew

xn

1113868111386811138681113868 m1113872 11138731113960 1113961

(10)

during the maximization or M-step -e new Θ and σ2parameters are obtained by rewriting equation (10)

Q Θ σ21113872 1113873 12σ2

1113944

N

n11113944

M+1

m1Pold

m | xn( 1113857 xn minus T ym θ( 1113857

2

+DNp

2log σ2

(11)

where D corresponds to data dimensionality -e algorithmalternatively iterates E-step and M-step until Qrsquos conver-gence thus aligning the sets of points with X T(YΘ)

However assigning the same probability P(m) (1M)

to all GMM components can affect registration quality given

that each Gaussian contributes differently in the model Inthis work it is proposed to weight the membership prob-abilities with the values obtained from the posterior prob-ability resulting from the RF classification process At thesame time the fetal cranium classification helps to obtain thepoint clouds used in the registration

In this way the GMM membership probabilities P(m)

correspond to the posterior probability (normalized between0 and 1) obtained during the classification stepP(c +1 | xn) (equation (4)) (where c +1 is the classcorresponding to fetal cranium) for each point xn It can beexpressed by

P(m)lowast

P c +1 | xn( 1113857

λ (12)

where

λ 1113944M

k1P c +1 | xn( 1113857k (13)

-erefore Pold(m | xn) and the Q objective function canbe rewritten as follows

Poldlowast

m | xn( 1113857 P c +1 | xn( 1113857exp minus xn minus T ym θ( 1113857

22σ21113874 11138751113882 1113883

1113936Mk1P c +1 | xn( 1113857k exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M

Qlowast

Q + 1113944N

n11113944

M

m1Pold

m | xn( 1113857logP(m)lowast

(14)

With this weighting strategy outliers will have less contributioncompared to voxels that describe the fetal cranium which willhave a larger weight during the registration process -esechanges do not affect the previously described EM optimi-zation procedure -us the Probabilistic Learning CoherentPoint Drift (PL-CPD) method has the advantages of being lesssensitive to outliers and do not requiring a high computationalcost compared to other intensity-based methods

24 Evaluation of the PL-CPD Method To assess RFrsquos per-formance in head segmentation a 5-fold crossvalidation wascarried out where the training and test data sets were ob-tained from a sample of five US volumes (an example of theclasses is presented in Figure 2) As segmentation perfor-mance metric the Dice index was computed which is de-fined as follows

Dice 2|AcapB|

|A| +|B| (15)

where A is the volume segmented by an expert in obstetricsand B represents the automatically segmented volumeAdditionally the Hausdorff Surface Distance (HSD) [39]and the area under the receiver operating characteristic

(ROC) curve (AUC) were obtained After segmentationvalidation the RF was applied to the remaining studies

Subsequently to validate the proposed PL-CPDmethodthe Target Registration Error (TRE) (equation (16)) betweenthe reference volume VR and the study to be aligned VM

were measured considering the eighteen pairs of registeredvolumes A total of n 1000 uniformly distributed targetpoints were defined as ai and bi corresponding to theregistered and reference volumes respectively

TRE VR VM( 1113857

1n

1113944

n

i1bi minus ai( 1113857

2

11139741113972

(16)

According to Fitzpatrick [40] TRE provides a betterestimation of registration errors than a landmark-basedmetric Additionally to evaluate in detail translation androtation errors the Root Mean Square (RMS) difference wasused It can be denoted as follows

RMS VR VM( 1113857

1m

1113944

m

k1bk minus ak( 1113857

2

11139741113972

(17)

To measure displacement errors the values in the threeaxes (ΔX ΔY and ΔZ) are considered as bk and ak with

6 Computational and Mathematical Methods in Medicine

m 3 on the other hand for the rotation errors the values ofthe angles (α δ and λ) are taken as bk and ak

To evaluate its performance the proposed PL-CPD al-gorithm was compared with other previously reported in-tensity-based registration methods Mutual Information(MI) Cross Correlation (CC) and Mean Squared Error(MSE) [6ndash8] -e method was also compared with somegeometric registration methods based on correspondencesuch as Iterative Closest Point (ICP) [12 13] RandomSample Consensus (RANSAC) [14] Scale-Invariant FeatureTransform (SIFT) [15] and Coherent Point Drift (CPD)[16] For all comparisons TRErsquos mean and standard devi-ation were obtained from the accumulated errors measuredfor each registration case Statistical differences for thesetests were determined by an paired Student t-test (plt 005)comparing the PL-CPD method with each other In addi-tion the mean RMS errors for translations and rotationswere obtained for each carried out registration

3 Results and Discussion

In this section results of the proposed registration method(PL-CPD) as well as the corresponding discussion arepresented As mentioned in Section 22 the procedure be-gins with a preprocessing step of background normalizationwith confidence maps An example of the results obtained isshown in Figure 3 fetal head image in axial view(Figure 3(a)) its corresponding confidence map(Figure 3(b)) and the image obtained after normalization(Figure 3(c)) It can be noted that the weighted image showsless intensity variation specifically in those regions with thelowest contribution of the confidence map (yellow andorange areas) As a result intensity variations around thefetal head are attenuated as can be seen by comparing theimages before and after confidence map weighting

To evaluate the performance of RF classification fivelabeled fetal head volumes were used where the task was todiscriminate between voxels corresponding to the fetalcranium (class c +1) and the rest of structures (classc minus 1) For this 5-fold crossvalidation was used Seg-mentation results for the five evaluated volumes are pre-sented in Table 1 It can be seen that all metrics showconsistent results with standard deviations of 35 1 and34 for the Dice HSD and AUC indices respectively Inaddition the average performances obtained by Dice andAUC metrics are higher than 80 which is consideredacceptable to generate the point clouds and the probabilisticmaps needed in the registration process

In Figure 4(a) an example of fetal cranium segmentationis presented where it can be seen that it contains severalmissing regions due to US acoustic shadows As alreadymentioned these missing data hinder the alignment processIn addition Figure 4(b) shows an example of the same pointcloud with the weighting factors obtained using RF It can benoticed that the distribution of weights is not uniform dueto the fact that not all the points have the same probability ofbelonging to the fetal cranium As detailed in the meth-odology section these weights are used as membershipprobabilities of the GMM used in the CPD method

-e visual outcome of each methodrsquos registration isshown in Figure 5 Volumes to be aligned were acquired inan axial and coronal view (Figures 5(a) and 5(b) respec-tively) To better appreciate registration results the ref-erence volume is shown in red (axial view) and the alignedvolume is presented in blue (coronal view) In all casesbecause both registered volumes contain different andcomplementary information differences in the visual re-sults of the registration are shown However a goodalignment result can be seen in structures such as cranialcircumference and ventricles In the second row resultsobtained with intensity-based methods are shown wherethe best performance corresponded to the MI algorithm-ese same methods combined with a binary mask pro-vided the registrations presented in the third row(Figures 5(f ) and 5(h)) It can be noticed that MI and CCimproved when using the binary mask contrary to MSE-e fourth row of Figure 5 shows the results of geometric-based methods where RANSAC presented the best per-formance observed with a better adjustment of the fetalcranial circumference Finally in the last row a compar-ison of the CPD method (Figure 5(l)) with the proposedmethod (PL-CPD) (Figure 5(m)) can be appreciated It canbe noted that performance measured by TRE of PL-CPD isbetter compared to CPD -e corresponding registeredimages show a good alignment of fetal heads and of thecerebral ventriclesrsquo central region

Visual results of the registered point clouds using CPDand PL-CPD methods can be seen in Figure 6 in red thefixed set of points and in blue the aligned points cloud It canbe observed that the PL-CPD method shows a slight im-provement in aligning both studies (Figure 6(b)) comparedto the CPD method (Figure 6(a)) -is can be due to thecontribution of the nonuniform assignment membershipprobabilities of the GMM

To perform a quantitative evaluation of the obtainedresults as well as a comparison with other registrationmethods commonly used in the literature the TRE wascalculated between the reference volume and the study to beregistered-e results are shown in Table 2 To determine thestatistically significant differences between the proposed andthe other methods a paired Student t-test (plt 005) wascarried out for each comparison -e first section of Table 2shows the results in millimeters of several intensity-basedalgorithms where MI presented the smallest error followedby MSE and CC all of them showed statistical differencescompared with the proposed method As expected thesemethods based entirely on intensity information presentconsiderable errors when aligning US fetal brain studies withacoustic shadows and artefacts that alter the registrationresult

As described in the methodology section the intensity-based methods were combined with a binary mask productof fetal head segmentation In this way registration wascarried out only with visible information without consid-ering acoustic shadows -e results are shown in the secondsection of Table 2 A general improvement can be observedin each method obtaining a 78 TRE reduction for MI-BM81 for CC-BM and 92 forMSE-BM compared with their

Computational and Mathematical Methods in Medicine 7

corresponding versions without using a binary mask De-spite these improvements these three methods presentstatistical differences with respect to the proposed methodBinary masks helped to attenuate artefacts introduced byacoustic shadows in the registration process but becauseboth registered studies do not have the same informationthe alignment result is affected

In the last section of Table 2 results obtained with geo-metric-based methods are presented of which ICP(978plusmn 465mm) SIFT (1028plusmn 483mm) and RANSAC(773plusmn 457mm) show statistical differences compared to PL-CPD In contrast CPD presents a small error with689plusmn 408mm Finally the result obtained with the proposedmethod shows the best performance of all with638plusmn 324mm which outperforms the CPD algorithm -isindicates that the incorporation of probabilistic weights asprobability of belonging contributes to the alignment process

Additionally computing times for each of the comparedalgorithms were measured and are shown in the last column

of Table 2 -e first section shows that intensity-basedmethods present the highest times with 375 plusmn 93min forMI 499 plusmn 132min for CC and 396 plusmn 161min for MSEcomputations When binary masks are incorporated to thesealgorithms times are considerably reduced down to 567519 and 566 for MI-BM CC-BM and MSE-BM re-spectively In contrast computation times for geometrymethods are inferior to those obtained by intensity-basedstrategies being 164 plusmn 88min for RANSAC followed bySIFT with 143 plusmn 62min -e PL-CPD algorithm shows anaverage computational time of 127 plusmn 48min which repre-sents an increase of 47 with respect to CPD Finally ICPtook the best computation time with 95 plusmn 27min but it alsopresents considerable registration errors-ese results concurwith a similar evaluation reported by Savva et al [11] where itis concluded that intensity methods are heavier in compu-tational cost because of a wider search space for optimalregistration Another aspect to be considered is that several ofthe compared correspondence-based and intensity-basedmethods combined with binary mask (marked with + inTable 2) require a previous fetal cranium segmentation withRF which adds 74 plusmn 26min to overall computation timesFurthermore given that RF is a supervised classifier it re-quires a previous training step (described in Section 22) thatrepresents an average time of 128 plusmn 34min -is step iscarried out only one and therefore it is not included incomputational times reported in Table 2

In a preliminary study an SVM classifier to segment fetalcranium was proposed -e TRE obtained by SVM was621 plusmn 378mm which is similar to the TRE shown by RF

250

200

150

100

50

0Grayscale

(a)

Confidencefactor

1

08

06

04

02

0

(b)250

200

150

100

50

0Grayscale

(c)

Figure 3 Example of the US imagesrsquo weighting using confidence maps (a) representative fetal head US image in axial view (b) cor-responding confidence map and (c) weighted image

Table 1 Quantitative evaluation of fetal head segmentation usingRF classifier

No Dice () HSD (mm) AUC ()1 913 48 8822 867 57 8243 886 51 8544 854 66 8355 82 73 792Global (μ plusmn σ) 868 plusmn 35 59 plusmn 1 837 plusmn 34

8 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

-e ldquonewrdquo parameter values are estimated byminimizingexpectation of the complete negative log-likelihoodfunction

Q minus 1113944N

n11113944

M+1

m1Pold

m | xn( 1113857log Pnew

(m)pnew

xn

1113868111386811138681113868 m1113872 11138731113960 1113961

(10)

during the maximization or M-step -e new Θ and σ2parameters are obtained by rewriting equation (10)

Q Θ σ21113872 1113873 12σ2

1113944

N

n11113944

M+1

m1Pold

m | xn( 1113857 xn minus T ym θ( 1113857

2

+DNp

2log σ2

(11)

where D corresponds to data dimensionality -e algorithmalternatively iterates E-step and M-step until Qrsquos conver-gence thus aligning the sets of points with X T(YΘ)

However assigning the same probability P(m) (1M)

to all GMM components can affect registration quality given

that each Gaussian contributes differently in the model Inthis work it is proposed to weight the membership prob-abilities with the values obtained from the posterior prob-ability resulting from the RF classification process At thesame time the fetal cranium classification helps to obtain thepoint clouds used in the registration

In this way the GMM membership probabilities P(m)

correspond to the posterior probability (normalized between0 and 1) obtained during the classification stepP(c +1 | xn) (equation (4)) (where c +1 is the classcorresponding to fetal cranium) for each point xn It can beexpressed by

P(m)lowast

P c +1 | xn( 1113857

λ (12)

where

λ 1113944M

k1P c +1 | xn( 1113857k (13)

-erefore Pold(m | xn) and the Q objective function canbe rewritten as follows

Poldlowast

m | xn( 1113857 P c +1 | xn( 1113857exp minus xn minus T ym θ( 1113857

22σ21113874 11138751113882 1113883

1113936Mk1P c +1 | xn( 1113857k exp minus xn minus T ym θ( 1113857

11138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386822σ21113872 11138731113966 1113967 + ωM 2πσ2( )

D2(1 minus ω)N1113872 1113873 m 1 2 M

Qlowast

Q + 1113944N

n11113944

M

m1Pold

m | xn( 1113857logP(m)lowast

(14)

With this weighting strategy outliers will have less contributioncompared to voxels that describe the fetal cranium which willhave a larger weight during the registration process -esechanges do not affect the previously described EM optimi-zation procedure -us the Probabilistic Learning CoherentPoint Drift (PL-CPD) method has the advantages of being lesssensitive to outliers and do not requiring a high computationalcost compared to other intensity-based methods

24 Evaluation of the PL-CPD Method To assess RFrsquos per-formance in head segmentation a 5-fold crossvalidation wascarried out where the training and test data sets were ob-tained from a sample of five US volumes (an example of theclasses is presented in Figure 2) As segmentation perfor-mance metric the Dice index was computed which is de-fined as follows

Dice 2|AcapB|

|A| +|B| (15)

where A is the volume segmented by an expert in obstetricsand B represents the automatically segmented volumeAdditionally the Hausdorff Surface Distance (HSD) [39]and the area under the receiver operating characteristic

(ROC) curve (AUC) were obtained After segmentationvalidation the RF was applied to the remaining studies

Subsequently to validate the proposed PL-CPDmethodthe Target Registration Error (TRE) (equation (16)) betweenthe reference volume VR and the study to be aligned VM

were measured considering the eighteen pairs of registeredvolumes A total of n 1000 uniformly distributed targetpoints were defined as ai and bi corresponding to theregistered and reference volumes respectively

TRE VR VM( 1113857

1n

1113944

n

i1bi minus ai( 1113857

2

11139741113972

(16)

According to Fitzpatrick [40] TRE provides a betterestimation of registration errors than a landmark-basedmetric Additionally to evaluate in detail translation androtation errors the Root Mean Square (RMS) difference wasused It can be denoted as follows

RMS VR VM( 1113857

1m

1113944

m

k1bk minus ak( 1113857

2

11139741113972

(17)

To measure displacement errors the values in the threeaxes (ΔX ΔY and ΔZ) are considered as bk and ak with

6 Computational and Mathematical Methods in Medicine

m 3 on the other hand for the rotation errors the values ofthe angles (α δ and λ) are taken as bk and ak

To evaluate its performance the proposed PL-CPD al-gorithm was compared with other previously reported in-tensity-based registration methods Mutual Information(MI) Cross Correlation (CC) and Mean Squared Error(MSE) [6ndash8] -e method was also compared with somegeometric registration methods based on correspondencesuch as Iterative Closest Point (ICP) [12 13] RandomSample Consensus (RANSAC) [14] Scale-Invariant FeatureTransform (SIFT) [15] and Coherent Point Drift (CPD)[16] For all comparisons TRErsquos mean and standard devi-ation were obtained from the accumulated errors measuredfor each registration case Statistical differences for thesetests were determined by an paired Student t-test (plt 005)comparing the PL-CPD method with each other In addi-tion the mean RMS errors for translations and rotationswere obtained for each carried out registration

3 Results and Discussion

In this section results of the proposed registration method(PL-CPD) as well as the corresponding discussion arepresented As mentioned in Section 22 the procedure be-gins with a preprocessing step of background normalizationwith confidence maps An example of the results obtained isshown in Figure 3 fetal head image in axial view(Figure 3(a)) its corresponding confidence map(Figure 3(b)) and the image obtained after normalization(Figure 3(c)) It can be noted that the weighted image showsless intensity variation specifically in those regions with thelowest contribution of the confidence map (yellow andorange areas) As a result intensity variations around thefetal head are attenuated as can be seen by comparing theimages before and after confidence map weighting

To evaluate the performance of RF classification fivelabeled fetal head volumes were used where the task was todiscriminate between voxels corresponding to the fetalcranium (class c +1) and the rest of structures (classc minus 1) For this 5-fold crossvalidation was used Seg-mentation results for the five evaluated volumes are pre-sented in Table 1 It can be seen that all metrics showconsistent results with standard deviations of 35 1 and34 for the Dice HSD and AUC indices respectively Inaddition the average performances obtained by Dice andAUC metrics are higher than 80 which is consideredacceptable to generate the point clouds and the probabilisticmaps needed in the registration process

In Figure 4(a) an example of fetal cranium segmentationis presented where it can be seen that it contains severalmissing regions due to US acoustic shadows As alreadymentioned these missing data hinder the alignment processIn addition Figure 4(b) shows an example of the same pointcloud with the weighting factors obtained using RF It can benoticed that the distribution of weights is not uniform dueto the fact that not all the points have the same probability ofbelonging to the fetal cranium As detailed in the meth-odology section these weights are used as membershipprobabilities of the GMM used in the CPD method

-e visual outcome of each methodrsquos registration isshown in Figure 5 Volumes to be aligned were acquired inan axial and coronal view (Figures 5(a) and 5(b) respec-tively) To better appreciate registration results the ref-erence volume is shown in red (axial view) and the alignedvolume is presented in blue (coronal view) In all casesbecause both registered volumes contain different andcomplementary information differences in the visual re-sults of the registration are shown However a goodalignment result can be seen in structures such as cranialcircumference and ventricles In the second row resultsobtained with intensity-based methods are shown wherethe best performance corresponded to the MI algorithm-ese same methods combined with a binary mask pro-vided the registrations presented in the third row(Figures 5(f ) and 5(h)) It can be noticed that MI and CCimproved when using the binary mask contrary to MSE-e fourth row of Figure 5 shows the results of geometric-based methods where RANSAC presented the best per-formance observed with a better adjustment of the fetalcranial circumference Finally in the last row a compar-ison of the CPD method (Figure 5(l)) with the proposedmethod (PL-CPD) (Figure 5(m)) can be appreciated It canbe noted that performance measured by TRE of PL-CPD isbetter compared to CPD -e corresponding registeredimages show a good alignment of fetal heads and of thecerebral ventriclesrsquo central region

Visual results of the registered point clouds using CPDand PL-CPD methods can be seen in Figure 6 in red thefixed set of points and in blue the aligned points cloud It canbe observed that the PL-CPD method shows a slight im-provement in aligning both studies (Figure 6(b)) comparedto the CPD method (Figure 6(a)) -is can be due to thecontribution of the nonuniform assignment membershipprobabilities of the GMM

To perform a quantitative evaluation of the obtainedresults as well as a comparison with other registrationmethods commonly used in the literature the TRE wascalculated between the reference volume and the study to beregistered-e results are shown in Table 2 To determine thestatistically significant differences between the proposed andthe other methods a paired Student t-test (plt 005) wascarried out for each comparison -e first section of Table 2shows the results in millimeters of several intensity-basedalgorithms where MI presented the smallest error followedby MSE and CC all of them showed statistical differencescompared with the proposed method As expected thesemethods based entirely on intensity information presentconsiderable errors when aligning US fetal brain studies withacoustic shadows and artefacts that alter the registrationresult

As described in the methodology section the intensity-based methods were combined with a binary mask productof fetal head segmentation In this way registration wascarried out only with visible information without consid-ering acoustic shadows -e results are shown in the secondsection of Table 2 A general improvement can be observedin each method obtaining a 78 TRE reduction for MI-BM81 for CC-BM and 92 forMSE-BM compared with their

Computational and Mathematical Methods in Medicine 7

corresponding versions without using a binary mask De-spite these improvements these three methods presentstatistical differences with respect to the proposed methodBinary masks helped to attenuate artefacts introduced byacoustic shadows in the registration process but becauseboth registered studies do not have the same informationthe alignment result is affected

In the last section of Table 2 results obtained with geo-metric-based methods are presented of which ICP(978plusmn 465mm) SIFT (1028plusmn 483mm) and RANSAC(773plusmn 457mm) show statistical differences compared to PL-CPD In contrast CPD presents a small error with689plusmn 408mm Finally the result obtained with the proposedmethod shows the best performance of all with638plusmn 324mm which outperforms the CPD algorithm -isindicates that the incorporation of probabilistic weights asprobability of belonging contributes to the alignment process

Additionally computing times for each of the comparedalgorithms were measured and are shown in the last column

of Table 2 -e first section shows that intensity-basedmethods present the highest times with 375 plusmn 93min forMI 499 plusmn 132min for CC and 396 plusmn 161min for MSEcomputations When binary masks are incorporated to thesealgorithms times are considerably reduced down to 567519 and 566 for MI-BM CC-BM and MSE-BM re-spectively In contrast computation times for geometrymethods are inferior to those obtained by intensity-basedstrategies being 164 plusmn 88min for RANSAC followed bySIFT with 143 plusmn 62min -e PL-CPD algorithm shows anaverage computational time of 127 plusmn 48min which repre-sents an increase of 47 with respect to CPD Finally ICPtook the best computation time with 95 plusmn 27min but it alsopresents considerable registration errors-ese results concurwith a similar evaluation reported by Savva et al [11] where itis concluded that intensity methods are heavier in compu-tational cost because of a wider search space for optimalregistration Another aspect to be considered is that several ofthe compared correspondence-based and intensity-basedmethods combined with binary mask (marked with + inTable 2) require a previous fetal cranium segmentation withRF which adds 74 plusmn 26min to overall computation timesFurthermore given that RF is a supervised classifier it re-quires a previous training step (described in Section 22) thatrepresents an average time of 128 plusmn 34min -is step iscarried out only one and therefore it is not included incomputational times reported in Table 2

In a preliminary study an SVM classifier to segment fetalcranium was proposed -e TRE obtained by SVM was621 plusmn 378mm which is similar to the TRE shown by RF

250

200

150

100

50

0Grayscale

(a)

Confidencefactor

1

08

06

04

02

0

(b)250

200

150

100

50

0Grayscale

(c)

Figure 3 Example of the US imagesrsquo weighting using confidence maps (a) representative fetal head US image in axial view (b) cor-responding confidence map and (c) weighted image

Table 1 Quantitative evaluation of fetal head segmentation usingRF classifier

No Dice () HSD (mm) AUC ()1 913 48 8822 867 57 8243 886 51 8544 854 66 8355 82 73 792Global (μ plusmn σ) 868 plusmn 35 59 plusmn 1 837 plusmn 34

8 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

m 3 on the other hand for the rotation errors the values ofthe angles (α δ and λ) are taken as bk and ak

To evaluate its performance the proposed PL-CPD al-gorithm was compared with other previously reported in-tensity-based registration methods Mutual Information(MI) Cross Correlation (CC) and Mean Squared Error(MSE) [6ndash8] -e method was also compared with somegeometric registration methods based on correspondencesuch as Iterative Closest Point (ICP) [12 13] RandomSample Consensus (RANSAC) [14] Scale-Invariant FeatureTransform (SIFT) [15] and Coherent Point Drift (CPD)[16] For all comparisons TRErsquos mean and standard devi-ation were obtained from the accumulated errors measuredfor each registration case Statistical differences for thesetests were determined by an paired Student t-test (plt 005)comparing the PL-CPD method with each other In addi-tion the mean RMS errors for translations and rotationswere obtained for each carried out registration

3 Results and Discussion

In this section results of the proposed registration method(PL-CPD) as well as the corresponding discussion arepresented As mentioned in Section 22 the procedure be-gins with a preprocessing step of background normalizationwith confidence maps An example of the results obtained isshown in Figure 3 fetal head image in axial view(Figure 3(a)) its corresponding confidence map(Figure 3(b)) and the image obtained after normalization(Figure 3(c)) It can be noted that the weighted image showsless intensity variation specifically in those regions with thelowest contribution of the confidence map (yellow andorange areas) As a result intensity variations around thefetal head are attenuated as can be seen by comparing theimages before and after confidence map weighting

To evaluate the performance of RF classification fivelabeled fetal head volumes were used where the task was todiscriminate between voxels corresponding to the fetalcranium (class c +1) and the rest of structures (classc minus 1) For this 5-fold crossvalidation was used Seg-mentation results for the five evaluated volumes are pre-sented in Table 1 It can be seen that all metrics showconsistent results with standard deviations of 35 1 and34 for the Dice HSD and AUC indices respectively Inaddition the average performances obtained by Dice andAUC metrics are higher than 80 which is consideredacceptable to generate the point clouds and the probabilisticmaps needed in the registration process

In Figure 4(a) an example of fetal cranium segmentationis presented where it can be seen that it contains severalmissing regions due to US acoustic shadows As alreadymentioned these missing data hinder the alignment processIn addition Figure 4(b) shows an example of the same pointcloud with the weighting factors obtained using RF It can benoticed that the distribution of weights is not uniform dueto the fact that not all the points have the same probability ofbelonging to the fetal cranium As detailed in the meth-odology section these weights are used as membershipprobabilities of the GMM used in the CPD method

-e visual outcome of each methodrsquos registration isshown in Figure 5 Volumes to be aligned were acquired inan axial and coronal view (Figures 5(a) and 5(b) respec-tively) To better appreciate registration results the ref-erence volume is shown in red (axial view) and the alignedvolume is presented in blue (coronal view) In all casesbecause both registered volumes contain different andcomplementary information differences in the visual re-sults of the registration are shown However a goodalignment result can be seen in structures such as cranialcircumference and ventricles In the second row resultsobtained with intensity-based methods are shown wherethe best performance corresponded to the MI algorithm-ese same methods combined with a binary mask pro-vided the registrations presented in the third row(Figures 5(f ) and 5(h)) It can be noticed that MI and CCimproved when using the binary mask contrary to MSE-e fourth row of Figure 5 shows the results of geometric-based methods where RANSAC presented the best per-formance observed with a better adjustment of the fetalcranial circumference Finally in the last row a compar-ison of the CPD method (Figure 5(l)) with the proposedmethod (PL-CPD) (Figure 5(m)) can be appreciated It canbe noted that performance measured by TRE of PL-CPD isbetter compared to CPD -e corresponding registeredimages show a good alignment of fetal heads and of thecerebral ventriclesrsquo central region

Visual results of the registered point clouds using CPDand PL-CPD methods can be seen in Figure 6 in red thefixed set of points and in blue the aligned points cloud It canbe observed that the PL-CPD method shows a slight im-provement in aligning both studies (Figure 6(b)) comparedto the CPD method (Figure 6(a)) -is can be due to thecontribution of the nonuniform assignment membershipprobabilities of the GMM

To perform a quantitative evaluation of the obtainedresults as well as a comparison with other registrationmethods commonly used in the literature the TRE wascalculated between the reference volume and the study to beregistered-e results are shown in Table 2 To determine thestatistically significant differences between the proposed andthe other methods a paired Student t-test (plt 005) wascarried out for each comparison -e first section of Table 2shows the results in millimeters of several intensity-basedalgorithms where MI presented the smallest error followedby MSE and CC all of them showed statistical differencescompared with the proposed method As expected thesemethods based entirely on intensity information presentconsiderable errors when aligning US fetal brain studies withacoustic shadows and artefacts that alter the registrationresult

As described in the methodology section the intensity-based methods were combined with a binary mask productof fetal head segmentation In this way registration wascarried out only with visible information without consid-ering acoustic shadows -e results are shown in the secondsection of Table 2 A general improvement can be observedin each method obtaining a 78 TRE reduction for MI-BM81 for CC-BM and 92 forMSE-BM compared with their

Computational and Mathematical Methods in Medicine 7

corresponding versions without using a binary mask De-spite these improvements these three methods presentstatistical differences with respect to the proposed methodBinary masks helped to attenuate artefacts introduced byacoustic shadows in the registration process but becauseboth registered studies do not have the same informationthe alignment result is affected

In the last section of Table 2 results obtained with geo-metric-based methods are presented of which ICP(978plusmn 465mm) SIFT (1028plusmn 483mm) and RANSAC(773plusmn 457mm) show statistical differences compared to PL-CPD In contrast CPD presents a small error with689plusmn 408mm Finally the result obtained with the proposedmethod shows the best performance of all with638plusmn 324mm which outperforms the CPD algorithm -isindicates that the incorporation of probabilistic weights asprobability of belonging contributes to the alignment process

Additionally computing times for each of the comparedalgorithms were measured and are shown in the last column

of Table 2 -e first section shows that intensity-basedmethods present the highest times with 375 plusmn 93min forMI 499 plusmn 132min for CC and 396 plusmn 161min for MSEcomputations When binary masks are incorporated to thesealgorithms times are considerably reduced down to 567519 and 566 for MI-BM CC-BM and MSE-BM re-spectively In contrast computation times for geometrymethods are inferior to those obtained by intensity-basedstrategies being 164 plusmn 88min for RANSAC followed bySIFT with 143 plusmn 62min -e PL-CPD algorithm shows anaverage computational time of 127 plusmn 48min which repre-sents an increase of 47 with respect to CPD Finally ICPtook the best computation time with 95 plusmn 27min but it alsopresents considerable registration errors-ese results concurwith a similar evaluation reported by Savva et al [11] where itis concluded that intensity methods are heavier in compu-tational cost because of a wider search space for optimalregistration Another aspect to be considered is that several ofthe compared correspondence-based and intensity-basedmethods combined with binary mask (marked with + inTable 2) require a previous fetal cranium segmentation withRF which adds 74 plusmn 26min to overall computation timesFurthermore given that RF is a supervised classifier it re-quires a previous training step (described in Section 22) thatrepresents an average time of 128 plusmn 34min -is step iscarried out only one and therefore it is not included incomputational times reported in Table 2

In a preliminary study an SVM classifier to segment fetalcranium was proposed -e TRE obtained by SVM was621 plusmn 378mm which is similar to the TRE shown by RF

250

200

150

100

50

0Grayscale

(a)

Confidencefactor

1

08

06

04

02

0

(b)250

200

150

100

50

0Grayscale

(c)

Figure 3 Example of the US imagesrsquo weighting using confidence maps (a) representative fetal head US image in axial view (b) cor-responding confidence map and (c) weighted image

Table 1 Quantitative evaluation of fetal head segmentation usingRF classifier

No Dice () HSD (mm) AUC ()1 913 48 8822 867 57 8243 886 51 8544 854 66 8355 82 73 792Global (μ plusmn σ) 868 plusmn 35 59 plusmn 1 837 plusmn 34

8 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

corresponding versions without using a binary mask De-spite these improvements these three methods presentstatistical differences with respect to the proposed methodBinary masks helped to attenuate artefacts introduced byacoustic shadows in the registration process but becauseboth registered studies do not have the same informationthe alignment result is affected

In the last section of Table 2 results obtained with geo-metric-based methods are presented of which ICP(978plusmn 465mm) SIFT (1028plusmn 483mm) and RANSAC(773plusmn 457mm) show statistical differences compared to PL-CPD In contrast CPD presents a small error with689plusmn 408mm Finally the result obtained with the proposedmethod shows the best performance of all with638plusmn 324mm which outperforms the CPD algorithm -isindicates that the incorporation of probabilistic weights asprobability of belonging contributes to the alignment process

Additionally computing times for each of the comparedalgorithms were measured and are shown in the last column

of Table 2 -e first section shows that intensity-basedmethods present the highest times with 375 plusmn 93min forMI 499 plusmn 132min for CC and 396 plusmn 161min for MSEcomputations When binary masks are incorporated to thesealgorithms times are considerably reduced down to 567519 and 566 for MI-BM CC-BM and MSE-BM re-spectively In contrast computation times for geometrymethods are inferior to those obtained by intensity-basedstrategies being 164 plusmn 88min for RANSAC followed bySIFT with 143 plusmn 62min -e PL-CPD algorithm shows anaverage computational time of 127 plusmn 48min which repre-sents an increase of 47 with respect to CPD Finally ICPtook the best computation time with 95 plusmn 27min but it alsopresents considerable registration errors-ese results concurwith a similar evaluation reported by Savva et al [11] where itis concluded that intensity methods are heavier in compu-tational cost because of a wider search space for optimalregistration Another aspect to be considered is that several ofthe compared correspondence-based and intensity-basedmethods combined with binary mask (marked with + inTable 2) require a previous fetal cranium segmentation withRF which adds 74 plusmn 26min to overall computation timesFurthermore given that RF is a supervised classifier it re-quires a previous training step (described in Section 22) thatrepresents an average time of 128 plusmn 34min -is step iscarried out only one and therefore it is not included incomputational times reported in Table 2

In a preliminary study an SVM classifier to segment fetalcranium was proposed -e TRE obtained by SVM was621 plusmn 378mm which is similar to the TRE shown by RF

250

200

150

100

50

0Grayscale

(a)

Confidencefactor

1

08

06

04

02

0

(b)250

200

150

100

50

0Grayscale

(c)

Figure 3 Example of the US imagesrsquo weighting using confidence maps (a) representative fetal head US image in axial view (b) cor-responding confidence map and (c) weighted image

Table 1 Quantitative evaluation of fetal head segmentation usingRF classifier

No Dice () HSD (mm) AUC ()1 913 48 8822 867 57 8243 886 51 8544 854 66 8355 82 73 792Global (μ plusmn σ) 868 plusmn 35 59 plusmn 1 837 plusmn 34

8 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

however the average processing time by SVM is209 plusmn 48min which is 608 greater than RF

All evaluations were carried out in MATLAB2019a witha 6 Coretrade Intelreg i5-8500 processor at 41GHz and a DDR4memory at 26GHz of 4Gb

-e detailed translation and rotation errors for each pairof registered studies can be seen in Figure 7 Comparison oftranslation errors obtained with the proposed PL-CPDmethod and intensity-based methods with and withoutbinary masks is presented in Figure 7(a) It can be noticedthat in all cases our method provides lower errors except inthe first case registration where MI-BM shows a slightlybetter result In contrast translations errors obtained withgeometric methods are shown in Figure 7(b) It can beobserved that the PL-CPD method presents better results inmost cases Only RANSAC algorithm exceeds PL-CPD inone of the alignments (case 9) and the CPD method exceedsPL-CPD in four cases regarding translation

On the other hand rotation errorsmeasured in degrees arepresented in Figures 7(c) and 7(d) In the former perfor-mances of intensity-based methods show that there is a highvariability -is phenomenon may be due to acoustic shadowsandmissing information on the studies It can be noted that inall cases the PL-CPD method is more stable and provideslower errors except for subject number six where it iscomparable to the MSE-BM approach When contrasting ourmethod with other geometry-based methods (Figure 7(d)) itcan be seen that rotation errors are competitive only out-performed in all cases by ICP and SIFT algorithms which isconsistent with what is reported in Table 2 ComparingRANSAC vs PL-CPD it can be seen that the proposed methodis exceeded in five pairs of volumes Finally when contrastingCPD vs PL-CPD methods it can be noticed that the latter issurpassed in only six pairs of alignments -is reflects that theproposedmethod incorporating results of RF voting process as

the GMM membership probability helps to attain a fineradjustment in the alignment process

Additionally it can be observed that the incorporation ofbinary masks boosted intensity-based methodsrsquo perfor-mance by only considering information of the segmentedhead during registration Also it must be noted that in casenumber four and seventeen the volumes presented severalocclusions which are reflected in higher rotation andtranslation errors for most of the tested methods In generalthe proposed method presents better results and less vari-ation compared to other intensity and geometry basedmethods

It should be considered that US volumes registered inthis research have a high percentage of occlusions noise andmissing data unlike applications previously reported byother authors where their data do not contain noise orocclusions On the other hand until now all reported var-iants of CPD use point clouds previously constructed Incontrast we propose an approach that builds the cloud ofpoints from the fetal cranium segmentation and at the sametime estimates the values of posterior probability incorpo-rated as probability weights of nonuniform membership ofthe GMM

In the context of aligning US data the errors obtainedfrom the proposed method are comparable with those re-ported by Fathima et al [29] who obtained a TRE of3plusmn 1mm in US studies However these authors did theregistration using 2D studies and without any artefacts-eymention that a gestational second-trimester fetus has acranial circumference going between 15ndash30 cm and abiparietal diameter of 45 to 6 cm therefore a TRE in therange of millimeters may be acceptable in clinical practiceOther works have carried out the registration of US fetalbrain studies in 2D and 3D However they have limitationssuch as registering in a single US study with itself and

(a)

1

08

06

04

02

0Weightfactor

(b)

Figure 4 Examples of fetal head point clouds models (a) set of points without weighting and (b) the same point cloud with probabilisticweights denoted by colors

Computational and Mathematical Methods in Medicine 9

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

without considering occlusion artifacts [27 28] or regis-tering phantoms where conditions are controlled and thereare no occluded areas [30]

As mentioned before unlike the previously reportedstudies the proposed method deals with the problem of US

brain volumes registration with a considerable degree ofoccluded areas and noise phenomenon that occurs in realcases mainly in second and third pregnancy trimesters Itconsiders that the proposed PL-CPD method may be usefulin the assessment of fetal growth to segment several internal

250

200

150

100

50

0GrayscaleAxial acquisition

(a)

250

200

150

100

50

0GrayscaleCoronal acquisition

(b)

250

200

150

100

50

0

Pixel intensities

Referenceimage

MITRE = 118mm

Registeredimage

(c)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CCTRE = 132mm

(d)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSETRE = 141mm

(e)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MI + binary maskTRE = 104mm

(f )

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CC + binary maskTRE = 128mm

(g)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

MSE + binary maskTRE = 172mm

(h)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

ICPTRE = 87mm

(i)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

SIFTTRE = 133mm

(j)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

RANSACTRE = 69mm

(k)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

CPDTRE = 53mm

(l)

250

200

150

100

50

0

Pixel intensities

Referenceimage

Registeredimage

PL-CPDTRE = 47mm

(m)

Figure 5 Representative example of each methodrsquos registration results using two volumes in axial (a) and coronal (b) view -e referencevolume (axial acquisition) is shown in red and the aligned volume (coronal acquisition) in blue

10 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

Table 2 TRE and computational times of different registration methods (μ plusmn σ)

Registration method TRE (mm) Computational time (min)Intensity-basedMutual information (MI) 1247 plusmn 41lowast 375 plusmn 93Cross correlation (CC) 1623 plusmn 677lowast 439 plusmn 132Mean square error (MSE) 1520 plusmn 753lowast 396 plusmn 161Intensity-based + binary maskMutual information (MI-BM) 974plusmn 403lowast 219plusmn 62+

Cross correlation (CC-BM) 1310plusmn 725lowast 228plusmn 41+

Mean square error (MSE-BM) 1398plusmn 731lowast 224plusmn 53+

Geometric-basedIterative closest point (ICP) 978plusmn 465lowast 95plusmn 27+

SIFT 1028plusmn 483lowast 143plusmn 62RANSAC 773plusmn 457lowast 164plusmn 88+

Coherent point drift (CPD) 689plusmn 408 121plusmn 51+

Probabilistic learning-CPD (PL-CPD) 638plusmn 324 127plusmn 48+

lowastStatistically significant differences with respect to PL-CPD method (plt 005) +-ese values include prior RF fetal cranium segmentation necessary togenerate the binary mask and the point cloud

CPDTRE = 53mm

(a)

PL-CPDTRE = 47mm

(b)

Figure 6 Representative example of the correspondence between a pair of aligned fetal head points cloud (a) Using the CPD method and(b) with the proposed method (PL-CPD)

10987654321RM

S tr

ansla

tion

erro

r (m

m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

0

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(a)

109876543210

RMS

tran

slatio

n er

ror (

mm

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(b)

Figure 7 Continued

Computational and Mathematical Methods in Medicine 11

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

brain structures or to combine different US fetal head ac-quisitions [4]

4 Conclusions

We presented a novel scheme for the automatic registrationof real 3D US fetal brain volumes volumes that contain aconsiderable degree of occlusion artefacts noise andmissing data To the best of our knowledge of the literaturethere are not any reported 3D alignment techniques usingreal fetal images with these characteristics

To register a new algorithm named PL-CPD is hereinproposed which incorporates two new features worthhighlighting automatic segmentation and conformation ofthe point cloud qualities that have not been reported by anypreviously developed CPD variant together with an ap-propriate proposed weighting of membership probabilitiesof the GMM -e segmentation and the weighting factorswere obtained from a supervised learning algorithm basedon a RF classifier and fed by a set of features composed ofintensity texture and edge patterns

-is weighting helps to attain a finer adjustment duringthe aligning of US data because not all points have the samemembership probabilities -e PL-CPD method can workwith point clouds contaminated with outliers missing dataor multiplicative noise these characteristics mean that PL-CPD can adequately register fetal brain US volumes even inthe presence of acoustic occlusions and speckle noise

When comparing the PL-CPD algorithm with otherintensity-based methods we obtained a better performancethis may be due to the fact that intensity-based methods areaffected by acoustic occlusions and therefore the studies tobe aligned do not share the same information whichcomplicated an adequate correspondence In comparisonwith other geometry-based methods PL-CPD preserves alower global error (TRE) and less deviation in translation

and rotation -is may be due to the incorporation ofnonuniform membership probabilities to the GMM

-e developed PL-CPD algorithm can be useful in clinicalpractice in one or all of the following cases to quantify fetalbrain growth and development to monitor the evolution ofindicators such as biparietal diameter or cranial circumfer-ence to segment different structures using a fetal braintemplate and to align and combinemultiple 3DUS fetal brainvolume acquisitions -e method can also be applied to othermedical image registration problems provided that theRandom Forest classifier can be adequately trained

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was partially funded by the Tec de MonterreyMIT Nanotechnology Program and the Programa de Apoyoa Proyectos de Investigacion e Innovacion Tecnologica(PAPIIT) IA102920 -e authors also acknowledge theNational Institute of Perinatology of Mexico (INPer) forsharing the Ultrasound images

References

[1] I E Timor A Monteagudo and H L Cohen ldquoNeuro-ecografia prenatal y neonatalrdquo vol 1 MARBANMexico CityMexico 2004

[2] International Society of Ultrasound in Obstetrics and Gy-necology ldquoSonographic examination of the fetal central

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDMICCMSE

MI-BMCC-BMMSE-BM

(c)

25

20

15

10

5

0

RMS

rota

tion

erro

r (deg)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18Registration number

PL-CPDICPSIFT

RANSACCPD

(d)

Figure 7 RMS registration errors of eachmethod In the first row translation errors of intensity and geometry-basedmethods are presented(a and b respectively) -e corresponding rotation errors are shown in the second row (c and d)

12 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

nervous system guidelines for performing the ldquobasic exam-inationrdquo and the ldquofetal neurosonogramrdquordquo Ultrasound inObstetrics amp Gynecology vol 29 no 1 pp 109ndash116 2007

[3] S M G V B Jardim and M A T Figueiredo ldquoSegmentationof fetal ultrasound imagesrdquoUltrasound inMedicine amp Biologyvol 31 no 2 pp 243ndash250 2005

[4] J Perez-Gonzalez F Arambula-Cosıo M Guzman et alldquoSpatial compounding of 3-D fetal brain ultrasound usingprobabilistic mapsrdquo Ultrasound in Medicine amp Biologyvol 44 no 1 pp 278ndash291 2018

[5] A Sotiras C Davatzikos and N Paragios ldquoDeformablemedical image registration a surveyrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1153ndash1190 2013

[6] F Maes D Vandermeulen and P Suetens ldquoMedical imageregistration using mutual informationrdquo Proceedings of theIEEE vol 91 no 10 pp 1699ndash1722 October 2003

[7] T Buzug and J Weese ldquoVoxel-based similarity measures formedical image registration in radiological diagnosis andimage guided surgeryrdquo Journal of Computing and InformationTechnology vol 6 no 2 pp 165ndash179 1998

[8] F Maes A Collignon D Vandermeulen G Marchal andP Suetens ldquoMultimodality image registration by maximiza-tion of mutual informationrdquo IEEE Transactions on MedicalImaging vol 16 no 2 pp 187ndash198 April 1997

[9] J F Krucker C R Meyer G L LeCarpentier J B Fowlkesand P L Carson ldquo3D Spatial compounding of ultrasoundimages using image-based nonrigid registrationrdquo Ultrasoundin Medicine amp Biology vol 26 no 9 pp 1475ndash1488 2000

[10] B Maiseli Y Gu and H Gao ldquoRecent developments andtrends in point set registration methodsrdquo Journal of VisualCommunication and Image Representation vol 46 pp 95ndash106 2017

[11] A D Savva T L Economopoulos and G K MatsopoulosldquoGeometry-based vs intensity-based medical image regis-tration a comparative study on 3D CT datardquo Computers inBiology and Medicine vol 69 pp 120ndash133 2016

[12] P J Besl and N D McKay ldquoA method for registration of 3-Dshapesrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 14 no 2 pp 239ndash256 1992

[13] Z Zhang ldquoIterative point matching for registration of free-form curves and surfacesrdquo International Journal of ComputerVision vol 13 no 2 pp 119ndash152 1994

[14] M A Fischler and R C Bolles ldquoRandom sample consensus aparadigm for model fitting with applications to image analysisand automated cartographyrdquo Communications of the ACMvol 24 no 6 pp 381ndash395 1981

[15] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60no 2 pp 91ndash110 2004

[16] A Myronenko and X Song ldquoPoint set registration coherentpoint driftrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 32 no 12 pp 2262ndash2275 2010

[17] P Wang P Wang Z Qu Y Gao and Z Shen ldquoA refinedcoherent point drift (CPD) algorithm for point set registra-tionrdquo Science China Information Sciences vol 54 no 12pp 2639ndash2646 2011

[18] Y Gao J Ma J Zhao J Tian and D Zhang ldquoA robust andoutlier-adaptive method for non-rigid point registrationrdquoPattern Analysis and Applications vol 17 no 2 pp 379ndash3882014

[19] S Liu G Sun Z Niu N Li and Z Chen ldquoRobust rigidcoherent point drift algorithm based on outlier suppressionand its application in image matchingrdquo Journal of AppliedRemote Sensing vol 9 no 1 Article ID 095085 2015

[20] S de Sousa and W G Kropatsch ldquoGraph-based point driftgraph centrality on the registration of point-setsrdquo PatternRecognition vol 48 no 2 pp 368ndash379 2015

[21] H Zhang W Ni W Yan J Wu and S Li ldquoRobust sar imageregistration based on edge matching and refined coherentpoint driftrdquo IEEE Geoscience and Remote Sensing Lettersvol 12 no 10 pp 2115ndash2119 2015

[22] M Lu J Zhao Y Guo and Y Ma ldquoAccelerated coherentpoint drift for automatic three-dimensional point cloudregistrationrdquo IEEE Geoscience and Remote Sensing Lettersvol 13 no 2 pp 162ndash166 2016

[23] P Zhang Y Qiao S Wang J Yang and Y Zhu ldquoA robustcoherent point drift approach based on rotation invariantshape contextrdquo Neurocomputing vol 219 pp 455ndash473 2017

[24] V Golyanik B Taetz G Reis and D Stricker ldquoExtendedcoherent point drift algorithm with correspondence priorsand optimal subsamplingrdquo in Proceedings of the 2016 IEEEWinter Conference on Applications of Computer Vision(WACV) pp 1ndash9 New York NY USA March 2016

[25] L Peng G Li M Xiao and L Xie ldquoRobust cpd algorithm fornon-rigid point set registration based on structure informa-tionrdquo PLoS One vol 11 no 2 Article ID e0148483 2016

[26] M Saval-Calvo J Azorin-Lopez A Fuster-Guillo V Villena-Martinez and R B Fisher ldquo3D non-rigid registration usingcolor color coherent point driftrdquo Computer Vision and ImageUnderstanding vol 169 pp 119ndash135 2018

[27] F Cen Y Jiang Z Zhang and H T Tsui ldquoShape and pixel-property based automatic affine registration between ultra-sound images of different fetal headrdquo in Lecture Notes inComputer Science pp 261ndash269 Springer Berlin Germany2004

[28] F Cen Y Jiang Z Zhang H T Tsui T K Lau and H XieldquoRobust registration of 3-D ultrasound images based onGabor filter and mean-shift methodrdquo Lecture Notes inComputer Science pp 304ndash316 Springer Berlin Germany2004

[29] S Fathima S Rueda P Aris and A Noble ldquoA novellocalndashphase method of automatic atlas construction in fetalultrasoundrdquo in Proceedings of the Medical Imaging 2011Image Processing Orlando FL USA March 2011

[30] H-C Chen P-Y Tsai H-H Huang et al ldquoRegistration-based segmentation of three-dimensional ultrasound imagesfor quantitative measurement of fetal craniofacial structurerdquoUltrasound in Medicine amp Biology vol 38 no 5 pp 811ndash8232012

[31] M Kuklisova-Murgasova A Cifor R Napolitano et alldquoRegistration of 3d fetal neurosonography and MRIrdquoMedicalImage Analysis vol 17 no 8 pp 1137ndash1150 2013

[32] C Che T S Mathai and J Galeotti ldquoUltrasound registrationa reviewrdquo Methods vol 115 pp 128ndash143 2017

[33] J Perez F Arambula M Guzman et al ldquoUltrasound fetalbrain registration using weighted coherent point driftrdquo inProceedings of the 12th International Symposium on MedicalInformation Processing and Analysis SPIE 10160 WashingtonDC USA January 2017

[34] A Karamalis W Wein T Klein and N Navab ldquoUltrasoundconfidence maps using random walksrdquo Medical ImageAnalysis vol 16 no 6 pp 1101ndash1112 2012

[35] A Criminisi and J Shotton Decision Forests for ComputerVision and Medical Image Analysis Springer PublishingCompany Incorporated Berlin Germany 2013

[36] R Gonzalez Digital Image Processing Prentice-Hall IncUpper Saddle River NJ USA 2002

Computational and Mathematical Methods in Medicine 13

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine

[37] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol PAMI-8 no 6 pp 679ndash698 1986

[38] Q Fang and D A Boas ldquoTetrahedral mesh generation fromvolumetric binary and grayscale imagesrdquo in Proceedings of the2009 IEEE International Symposium on Biomedical ImagingFrom Nano to Macro pp 1142ndash1145 Boston MA USA June2009

[39] F Prados J Ashburner C Blaiotta et al ldquoSpinal cord greymatter segmentation challengerdquo NeuroImage vol 152pp 312ndash329 2017

[40] J Fitzpatrick ldquoFiducial registration error and target regis-tration error are uncorrelatedrdquo in Proceedings of the MedicalImaging 2009 Visualization Image-Guided Procedures andModeling SPIE 7261 Orlando FL USA February 2009

14 Computational and Mathematical Methods in Medicine