A 3D MRI Skull Segmentation Method Based on DeformableModels.

Miguel Salas, Steve MaddockDepartment of Computer Science, University of She¢ eld

211 Portobello Street, She¢ eld, S1 4DP, UK{salasmig,S.Maddock}@dcs.shef.ac.uk

December 22, 2009

Abstract

The aim of this work is to extract the outer skullsurface from an MRI volume. Based on a 3D ap-proach, the technique proposed takes into accountthe information of the skull contained in MRI vol-umes of the head. Our main interest in extractingthe skull from MRI data is to create models of thehead in order to create a database of models wherethe relationship between the skull and face can bestudied. The analysis of this dependence will sup-port forensic craniofacial reconstruction methods inproducing more reliable face estimations.The segmentation technique proposed makes use

of an explicit 3D deformable model of a skull evolv-ing at each iteration considering two main aspects:volume features and skull shape restrictions mod-elled from statistical data.Index Terms: mri skull segmentaion, deformablemodels, shape analysis.

1 Introduction

In forensic applications the relationship betweenthe skull and face is a key aspect in creating fa-cial models to identify missing people from theirskull remains [29] [28] [31], especially when othermethods for identi�cation cannot be applied. Atthe present time, facial reconstruction techniquesrely on the hypothesis that the shape of the faceis de�ned in a straightforward way by the shape ofthe underlying skull.This skull-face relationship is usually condensed

in the form of anthropometric tables, giving tis-

sue depth measurements at a discrete set of pointsdistributed in prominent areas of the skull andface (anatomical landmarks). These tables, to-gether with personal experience, are used by foren-sic artists to produce facial reconstructions. Ourresearch work is a �rst step towards creating morecomprehensive sets of data for creating 3D skull-face models, thus improving the existing sourcesof information for supporting forensic facial recon-struction applications.

In this paper a novel method for segmentingskull in MRIs is presented that incorporates sta-tistical knowledge of the skull into a deformablemodel. Our data source is a collection of MRIvolumes of the head from which we segment thebone. First, candidate bone areas are separatedfrom background pixels and other tissue regionsbased on their intensity values. Then, these re-gions are re�ned using a deformable model whichevolves covering these regions according to a setof constraints imposed by the image features andskull shape statistics. The image features used cor-respond to a gradient vector �ow �eld (GVF) de-�ned on the initial volume to segment. On theother hand, the skin layer is extracted from theMRI data using the marching cubes algorithm [19].The results are a set of 3D skull-face models wherethe tissue depth is given at any point of the skullsurface.

The rest of this report is organised as follows: InSection 2 the previous work and main approachesrelated to the problem of skull segmentation in MRIare described. Section 3 presents the Bayesian ap-proach for image segmentation used in this research

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work. The key aspects related to representationand shape modelling are introduced in section 4.The 3D skull segmentation technique proposed andthe main results are presented in section 5. Finally,the conclusions of this work are presented in section6.

2 Previous work

In the area of image segmentation, previous workhas tried to extract the skull from MRI by con-sidering intensity homogeneity properties of boneregions. Mathematical Morphology is an spatialtechnique based on set theory applied to imageanalysis providing a quantitative description of geo-metrical structures. This technique can provideboundaries of objects, their skeletons, and theirconvex hulls. The starting point of this techniqueis the selection of a set of voxels representing possi-ble skull regions. These regions are then processedwith morphological operators based on spandingand shrinking operations [12].Another type of algorithm commonly used to ex-

tract the information of medical images is basedon the analysis of separable patterns in the im-age. Techniques in this category, assume that vox-els possess speci�c homogeneous attributes allow-ing their classi�cation in terms of intensity, colour,texture or movement. With models of parame-terised distributions for each type of tissue, theyfocus to solve the partial volume problem [23] inorder to produce adequate classi�cations of pixels.When the information in the image is di¢ cult toclassify, deformable models approaches are used todeal with the problem of pixel classi�cation. In gen-eral the idea of deformable models is to incorporateadditional information to the information providedby the image features.Generally, mathematical morphology techniques

work well for segmenting areas of the skull wheresmooth variations occur in shape and topology.However, the conditions assumed in these morphol-ogy techniques related to the regularity of skull re-gions can only be guaranteed in the upper part ofthe skull (cranium), and for this reason, these al-gorithms produce acceptable results only in thatarea. Under this consideration the works of Salasand Succar [24], Jere [14] and Dogdas [9] proposemethods for extracting the skull based on a spa-

tial processing of the volume using mathematicalmorphology operators. These algorithms discardirrelevant regions by removing background voxels1

and classifying other separable types of tissues suchas the scalp and the brain.In contrast, the algorithm proposed in this re-

search produces models of the entire skull volume,accounting for the most probable con�guration ofthe skull regions provided by volume features.The approach proposed in our work takes into

account probable skull components of the MRI vol-ume in a holistic formulation, even if these compo-nents are not connected with each other. Addi-tionally, the skull models produced in our researchshare a common structure (referred to the sametriangular mesh structure) which facilitates locat-ing important features and conducting statisticalanalysis of the models.Whilst probabilistic approaches for classifying

tissue types have also been applied successfully tosegment body organs in medical images, their ap-plication for the problem of skull segmentation inMRIs has failed to produce acceptable results inthe frontal area of the skull. Examples of these ap-proaches are the work of Leemput et. al. [17, 18],Laindlaw [16] and Heinoen [13]. The main reasonfor the limited results is because the separabilityassumptions for di¤erentiate tissues are di¢ cult tomeet when presented with skull regions in MRI.The air and the skull voxels have practically nodi¤erence in intensity, colour or texture attributes.The only di¤erence between these two types of tis-sue is the spatial position of each voxel with respectto the global spatial structure the head. Figure 2.1illustrates this situation. Also, there exist regionsof the skull with high variations in their intensityvalues, especially in areas of the skull with highconcentrations of fat. In contrast to the intensityproperties of most of bony regions, fat produce vox-els with very high intensity levels. In summary, inthe case of the skull, the homogeneous intensity re-gion hypothesis may be strongly violated.In this respect, the technique we propose is espe-

cially formulated to deal with this type of problem.Our algorithm takes into account the spatial posi-tion of candidate skull voxels with respect to thestructure of the probable skull they are describing,

1As pixel refers to picture elements, voxel refers to volumeelements.

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integrating relevant voxels in a single structure.In our technique, during the segmentation

process, some of the candidate skull voxels may ormay not belong to the skull. To decide which voxelsare considered, our algorithm has a mechanism forcompensating missing parts of the skull but also re-moving outliers and noise according to a statisticalmodel.

Figure 2.1: Binary images showing two exampleswere traditional techniques of tissue classi�cationfail to correctly detect skull areas. (Left) The pic-ture elements in area (a) have exactly the sameintensity, colour and texture that picture elementsin regions (b) and (c). (Right). Even removing thebackground regions (ignoring (b) and its connectedneighbours), it is not possible to di¤erentiate be-tween skull and non skull picture elements from re-gions (a) and (c). However, our knowledge of the"shape of a skull" suggest us that there must be aborder in the neighbourhod of area (a).

Deformable methods guided purely by voxel in-tensities have also been used for MRI skull segme-nation. The work of Rifai [23, 22], Mang [20], andGhadimi [11] are examples of these techniques.The main problem with these techniques is that

when considering only intensity information forguiding the deformable model, it may be attractedto incorrect boundaries. In addition, the resultsare highly sensitive to the intitialisation of the de-formable model. These techniques also rely on sep-arability and regularity assumptions for the mate-rials to segment, assumptions previously discussedthat are di¢ cult to guarantee in the case of skullregions in MRI data.Our approach addresses such problems using two

strategies: First, a registration step matching fea-tures between a clean skull model and a rough noisy

volume to be segmented is used to initialise thedeformable model. Second, the deformable modelevolves at each step to incorporate the informationprovided by the image features (GVF �eld) and thestatistically derived shape term.The Shan�s mehtod [26] for segmenting the skull

belongs to another type of technique based on com-bining CT with MRI information to extract theskull. In Shan�s work, a set of skull models gener-ated from CT segmentations is used for segmentingskull data in MRIs. The method is used to modelthe spatial positions of the skull voxels in an MRIgiven the information provided by CT scans of thesame individual. Then this information is used toestimate skull regions in MRI of di¤erent individ-uals. In the work of Payan et. al. [21] a skullapproximation method is proposed to extract skullinformation from MRI images for its use in cranio-facial reconstructions. They use a set of statisticalmodels consisting of a sparse set of the skull-facepoints obtained from CT data. The main short-coming is that the estimation of the skull shapes ismodelled in terms of a limited set of points alongthe face combined with tissue depths at these spe-ci�c positions. They use this set of landmarks asanchor points in order to produce rough approxi-mations of new skull models embedded in MRI.The main drawback of these techniques is that

they require a number of CT scans in order to pro-duce the initial skull models. CT scans producea high radiation dose, and it can be harmful toscan healthy people. Another disadventage is thatthe segmentation produced is simply a collection ofisolated voxels that in a second stage have to beintegrated to create a skull model.In contrast, the technique proposed in our work

only requires MRI to extract the skull which makesthe technique suitable to collect more data of livepeople. MRI, an acquisition technique that is notharmful, provides very detailed information of sev-eral tissue types present in the head. Additionally,as mentioned before, the result of our research iscomplete skull models which can be considered asa high level representation. These representationsprovide �exible models with advantages such as thecapability to referring to speci�c anatomical partsof the skull in an explicit way (if instead of a trian-gular mesh, other primitives such as pixels, voxelsor point clouds were obtained, it would be neces-sary an additional step to group the primitives in

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meaningful objects).

3 A Bayesian Approach ForIncluding Prior Knowledge

In this research work, the skull is extracted from anMRI volume by �nding a set of 3D parameters of adeformable model M which minimise a functionalof the form:

E(M) = Evolume(f; V ) + Eshape(M) (1)

where the term Evolume is the energy contributionof a Gradient Vector Flow �eld de�ned in termsof volume features, and Eshape bene�ts model con-�gurations with similar shapes to the skull beingsegmented. This shape term contains informationextracted from a set of training shapes.The parameter >= 0 is used to adjust the

amount of in�uence of the shape term in the equa-tion.The term Evolume relates to how well the de-

formable model M segments an input volume V ,based on the intensity information provided by aset of volume features f . The image feature se-lection depends on a particular intensity thresholdvalue for grouping voxels with similar properties.This selection forms a volume that is used to de-�ne a vector �eld in its neighbour areas which isused to attract the deformable model towards itscontouring regions.The term Eshape, represents high-level knowl-

edge about the geometric properties of a skull whichis previously acquired in a learning stage.The e¤ect of combining Evolume and Eshape is

twofold. First, it augments the capture range ofpotential �eld forces, leading to an approach lesssensitive to initialisation. Second, it improves thecapacity of the deformable model to deal with oc-clusion problems by adding knowledge of the shapeof the object to segment.Minimising the energy in equation 1 is equiv-

alent to maximizing the Bayesian Inference termP (M=f) de�ned by:

P (M=f) =P (f=M)P (M)

P (f)(2)

which means optimising the probability of a con-�guration for a deformable model M given the ob-served volume features f (i.e. obtaining the modelM that is the most probable according to the in-formation in the image).Usually, the term P (M) de�ned in deformable

model techniques, is used for modelling that largesurfaces are less probable [5]:

P (M) _ exp(�� jM j)where j M j is a measure of the area of the modelM .However, as will be shown in a later section, in

this paper we will use for our purposes a more elab-orate shape dissimilarity measure de�ned with thefollowing formulation:

P (M j fMig)with this expression representing the probability ofthe current model M with respect to a set of mtraining shapes fMigi=1::m: In the next section, themain aspects related to representation and learningof shapes will be presented in more detail.

4 Shape Representation andLearning

In this research, an explicit model of a skull in theform of a triangle mesh is used as the deformablemodel.Formally, the model M is de�ned with a pair of

sets (V;T), where V is a set of vertices v 2 R3, Tis a set of triplets of edges T � f(u; v; w) j u; v; w 2E^u 6= v 6= wg de�ning triangular polygons, and Eis a set of edges between vertices E � f(s; t) j s; t 2

V^s 6= tg. The union of all the triangleskSi=1

ti 2 T

de�nes a continuous closed surface S(M).The shape s associated with the deformable

model M is a set of N three-dimensional controlpoints de�ning the shape of a triangle meshM withthe control points set being a subset of the set ofvertices V.For simplicity, the shape s of a skull model will

be represented as a vector of coordinates with thefollowing structure:

s = (x1; y1; z1:::; xN ; yN ; zN )T (3)

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with xi, yi, zi being the coordinates of each controlpoint. The number N of control points is �xedand the same for each corresponding shape. In ourcase we use a shape model with N = 3000 controlpoints distributed along the surface of the skull,using more points in the frontal area of the face.Given the high detail of the triangular mesh used, itis possible to obtain good approximations of normalvectors practically at every point of the deformablemodel.Under this convention, the energy of the de-

formable model M in terms of its shape descriptors is the following:

�(s) / exp(�12(s� �)T��1? (s� �));

and the energy term to minimise is:

Eshape(s) = log(�(s)) + const (4)

= �12(s� �)T��1? (s� �) (5)

in these expresions, � represents the averageshape of the training sets.

4.1 Metrics of the Shape Models

Given two deformable models Ms and Ms de�nedas presented in the previous subsection, a Taylorexpansion is used to approximate the distance met-ric between them:

kMs �Ms k2� min�

ZSm

(Ms �M�s)2 (6)

which accounts for all continuous an monotonousreparametrisations of the models. In this work anapproximation of the Mahalanobis distance is usedby a simple Euclidean distance d between the con-trol points of the polygons:

d(Ms;Ms) � (s� s)T (s� s) (7)

4.2 Alignment of Training Shapes

Given a set of m training vectors � = fsigi=1::mwhich are centered and normalised, we are inter-ested on �nding an optimal alignment for dealingwith the scale and pose estimation of the shapes[10], [8]. An optimal alignment of two shapes s and

s with respect to rotations, translation and scal-ing (known as full Procrustes �t [10]) requires thefollowing distance to be minimsed:

D2(s; s) =k s� �s�� 1k T k2 (8)

where D is the distance between the two shapes,� >= 0 is a scaling factor, � is a rotation ma-trix, 1k is a vector of ones (k x 1) vector, and avector accounting for translations. Setting the cor-responding derivatives to zero, the solution for theoptimal parameters �, , and � are the followingexpressions [8], [27] and [10]:

= 0 (9)

� = UV T (10)

The rotation term � is de�ned in terms of thematrices U and V derived from a single value de-composition of the matrix product sT s

kskksk as follows:

sT s

k s kk s k = V �UT (11)

It can be shown that the best rotation estimator� can be obtained by the following ratio:

� =trace(sT s�)

trace(sT s)(12)

and �nally, the expression de�ning the best align-ment for the shape s is:

s = �sc� + 1k T +

pD2(sc; s) (13)

where sc is the centered version of shape s.

4.2.1 Average Shape of the Training Set

In this research, we align the shapes of the trainingset with respect to the Procrustes estimate of themean vector which is de�ned as:

� = arg inf�:S(�)=1

nXi=1

sin2 �(si; �)

= arg inf�:S(�)=1

nXi=1

D2(si; �)

the point in shape space corresponding to thearithmetic mean of the Procrustes �ts,

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�s =1

n

nXi=1

sPi (14)

has the same shape as the full Procrustes mean [10].The superscript P is used to denote the procrustessuper superimposition of shape si.

4.3 Gaussian Model For Represent-ing Shapes

It is assumed that the training shapes are aligned asde�ned in the previous subsection and distributedaccording to a multivariate Gaussian distribution.A statistcal shape model based on PCA is pro-

posed to model the shape variability of a given con-�guration with respect to a set of trained shapes.Here, the model proposed by Cremers for shapelearning [6] is extended to 3 dimensions.Let � = fsi 2 R3Ngi=1::m be a set of training

shapes, aligned as presented in section 4.2 withmean vector described in 14. The sample covari-ance matrix is given by:

� =1

m� 1

mXi=1

(si � �s)(si � �s)T (15)

A principal component analysis can be appliedto this covariance matrix in order to obtain themain sources of variation in the training set. PCAis an orthogonal linear transformation that trans-forms the data to a new coordinate system suchthat the greatest variance by any projection of thedata comes to lie in the �rst coordinate (�rst com-ponent), the second greatest variance in the secondcoordinate and so on. PCA is the optimum trans-form for given data in least square terms [15].The matrix � can be diagonalised and a set of

�1:::�r eigenvalues can be obtained. The modesof largest variation, given by the vectors ei; corre-spond to the largest eigenvalues �i. A compactlower-dimensional shape model can be obtained bylinear combination of these eigenmodes added tothe mean shape:

s(�1:::�r) = �s+rXi=1

�ip�ei

where r < m:

The factorp� has been introduced for normal-

isation and corresponds to the standard deviationin the direction of the vector ei.In general if the number of sampled elements is

smaller than the dimension of the underlying vec-tor space (2N), the covariance matrix � will nothave a full rank and the probability density willnot be supported in the full 2N dimensional space.This situation represents a problem for evaluatingequation 4 for shape con�gurations out of the spacede�ned by the training set (the probability distrib-ution is unde�ned). To solve this problem, a tech-nique of covariance regularisation is applied. Thefollowing section presents a solution to this prob-lem.

4.3.1 Regularising the Covariance

We use an approximation to the covariance ma-trix in order to simplify the problem of invertibility.The covariance matrix is expressed as a decompo-sition in eigenvalues and eigenvectors having thefollowing structure:

� = V DV T (16)

where D is the diagonal matrix of non-zero eigen-values �1 � ::: � �r > 0, and V is the matrix ofcorresponding eigenvectors2 . The covariance ma-trix is regularized by replacing all the zero eigen-values by a constant �? > 0: With this strategy,the space of all the shapes not considered in thetraining set space are assigned a small probabilityvalue.Thus, the new regularized covariance �? is

formed replacing the original matrix D? by D :

�? = V D?VT (17)

D? = D + �?(I � eveTv ) (18)

where evis an orthonormal basis of the matrixV , and I is the identity matrix. For this work, assuggested in [7], �? is given by:

�? =�r2

(19)

This expression guarantees that every possible vari-ation in the shape space will have a corresponding

2Here we are using � to denote the sorted eigenvalues �:

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value of probability �(s) covered by the covari-ance matrix �?. Better yet, equation 4 will bedi¤erentiable on the full space, associating a �nitenon-zero value with any shape s. This property be-comes crucial when it is necessary to maximise theprobability.

4.3.2 Properties of the Shape SpaceSpanned

In general , the number of samples needed to obtainreliable statistics increases rapidly with the dimen-sion of the input data [2, 1]. The covariance reg-ularisation technique used in our research and de-scribed in section 4.3.1 is a key aspetct allowing toreduce the impact of having a small number of sam-ples. We use complete models of the skull to createthe deformable model however the training set canbe divided to use some portions of the data to workon especi�c areas of the skull. In general, the mainproperties of the shape representation used in ourapproach are:

� it is possible to focus on the low-dimensionalsubspace de�ned by the training data

� it is possible to assign probabilities to dataeven in orthogonal directions to the subspacespanned by the training data

� as long as the number of samples increase, itis expected to have more reliable estimates ofthe mean and covariance matrix.

4.3.3 Invariance of the Shape Term in 3D

Since the training shapes are aligned to the meanshape �, the energy term has to be calculated con-sidering translation, rotation and scaling (to makeit correspond to an aligned shape). Also, this termneed to be normalised to have a unitary size (sec-tion 4.2). On the other hand, the same process hasto be applied to the argument s before calculat-ing the energy term presented in equation (4). Theenergy for the aligned and centered shape s is:

Eshape(s) = �1

2(s� �)T��1? (s� �)

with s as de�ned in equation (13). The term scwhich represents the shape centered (i.e. withtranslation of the shape s eliminated) is obtainedby:

sc = (I3n �1

n�)s

with:

� =

0BBBBB@1 0 0 1 0 0 � � �0 1 0 0 1 0 � � �0 0 1 0 0 1 � � �1 0 0 1 0 0 � � �...

......

......

.... . .

1CCCCCA (20)

The energy can be minimised by applying thechain rule on the gradient descent equation:

ds

dt= �dEshape(s)

ds= �dEshape(s)

ds� dsdsc

� dscds

(21)

with:

dEshape(s)

ds= (��1? (s� �))T (22)

dscds

= (I3n �1

n�) (23)

ds

dsc=d(�sc� + 1k

T +pD2(sc; s))

dsc(24)

note that the terms D; ; �; and � in equation 24are all functions of the aligned shape s as expressedin equations 8-12.

5 3D segmentation Algorithm

5.1 Overview of the 3D Skull Ex-traction Process

This section describes the modules involved in theprocess of skull extraction from MRI data pro-posed. In general terms, this approach can be con-sidered as a bootstraping technique. From an ap-proximate description of the object of interest theaim is to build better approximations of this ob-ject iteratively, based on the initial con�gurationand the features found in the volume dataset.The pre-processing module receives as input the

initial MRI volume and creates a �rst approxima-tion of the skull volume together with a set of vol-ume features associated to it. These features are

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then processed and a gradient vector �eld is ob-tained from them. The initialisation of the de-formable model is made by registering the pre-segmented volume with an initial clean model of theskull. Registration of these two models is accom-plished by matching prominent features previouslyidenti�ed in these two models. Figure 5.1 showsthe initialisation of the deformable model using aset of 3D curves to direct the matching process.After this initialisation, the shape and image fea-

ture terms work together in order to control thetemplate deformation at each step of the algorithm.The initial approximation of the skull volume isprocessed and re�ned iteratively using volume fea-tures, the skull template and the statistical infor-mation about skull shapes provided by the statis-tical module. The pre-processing and statisticalprocessing modules work together with the 3D seg-mentation module in an iterative algorithm to pro-duce a 3D skull surface. A diagram illustrating theorganisation of the modules is shown in �gure 5.2.Figure 5.3 shows the �ow diagram of the iterative

process proposed to segment the skull shape. Thedeformable model is initialised using a non-rigidregistration algorithm [4] based on matching a setof curves in places of the skull with prominent cur-vatures. The noisy pre-segmented volume and theinitial template are then make coincide their cor-responding features. Figure 5.1 shows a schematicexample of this initialisation step.For each control point, we calculate the in�uence

of the image features and shape terms and storethese in directional vectors d1; d2 and d3 (takinginto account each coordinate of the points). If thesecontrol points are close enough (d1) to the noisyapproximation boundaries, a di¤erential quantityin the direction of the nearest point of the noisyskull surface is calculated. If the control point is notnear to the noisy volume (d2) then the informationof the GVF �eld is used to calculate a diferentialo¤set in the direction indicated by the vector �eldin that point.In a second stage, these control points are used

to deform the template considering the informationcalculated for vectors d1 and d2 and the shape re-strictions calculated from the whole set of controlpoints in their current state (calculating a displace-ment vector d3). The deformation function used isa Radial Basis Function - Thin Plate Spline (RBF-TPS) [3]. The model is deformed in small quan-

tities with a simulated annealing strategy in orderto give more priority to the image features in the�rst iterations and give more weight to the shapeterm in the �nal iterations. This was implementedwith this strategy in order to preserve the topo-logical properties of the deformable model mesh ateach step of the algorithm. Also, local processingis possible in areas of the skull where more detail isrequired to control correctly the shape term. Thelocal processing is also useful to account for possi-ble peculiarities of the model being segmented.

Figure 5.1: Feature lines are extracted from boththe noisy pre-segmented skull Ns and the cleanskull template To. This gives a set of correspon-dence features in form of surface curves, that canbe registered to de�ne the initial shape of the tem-plate. With this pair of features a warping processis de�ned using the relation between the surfacecurves resulting in the initial skull template Ti.

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Figure 5.2: Elements of the 3D Segmentation process

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Figure 5.3: Flow Chart of the 3D segmentation algorithm.

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5.2 Results of the 3D SegmentationApproach

Three main aspects are discussed in this section:

1. The behaviour of the segmentation algorithmwhen its shape term is changed and how thischange a¤ects the result of the algorithm

2. The results of the segmentation algorithmwhen the two main factors involved vary: theimage feature factor (IFF) and the shape fac-tor (SF)

3. The evaluation of the results in terms of shapequality, which means how near/far the resultis from an average shape in terms of the shapeenergy calculated between these models.

5.2.1 E¤ect of the Shape Term Variation

To assess the results of the implemented algorithm,an initial skull base model B was deformed using afree form deformation algorithm (FFD) [30]. Thedeformed modelD was created, modifying arbitrar-ily the control point positions in the model B. Themodel D is processed in order to deform it back toa valid skull model, by means of applying our algo-rithm with only the shape term acting on it. Foreach parameter combination i; the resulting skullmodels produced by the algorithm at each itera-tion j will be referred as Dj

i . A target model Twas created as a reference model (used for the com-parisons), by applying to the deformed model Dour shape recovering algorithm with a shape factor(SF ) of 1.0 for 100 iterations.Figure 5.4 shows the base (B) and deformed (D)

models used for this evaluation. Figure 5.5 showsthe di¤erence in mm between the source and targetmodels.The graph in �gure 5.6 shows the distances be-

tween Dji and the reference model T at each itera-

tion j. The graph illustrates the behaviour of thealgorithm for the parameters: (i = 1, SF = 0:1),(i = 2, SF = 1:0) and (i = 3, SF = 10:0) and 20iterations.The reason for varying the evolution of the de-

formable models in small amounts is because in thisway, properties of the deformable model, such asthe smoothness of the surface and the topology ofthe mesh, are maintained.

Figure 5.4: Original target model B (Left) and anarbitrarily deformed model D created by applyinga free form deformation (FFD) technique to themodel in the left.

Figure 5.5: Surface comparison between modelsin �gure 5.4 left and right. The image shows acoloured model in a BGYR colour scale. Blue rep-resents nearest points while red represents pointswith more that 20mm of di¤erence.

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Figure 5.6: Graph showing the behaviour of theshape term result with respect to a pre-establishedaverage model after 80 iterations.

Figure 5.7: Results of for the three shape parame-ters described in graph 5.6 i=1, factor=0.1(left),i=2, factor=1.0(middle), and i=3, factor=10.0(right) after 80 iterations.

From �gure 5.6 it can be observed that, withSF = 1:0, the resulting model converges towardsthe solution (i.e. perfect match has a distanced = 0:0) in a regular and smooth descending wayafter 30 iterations. Also, the graph shows that byreducing the shape factor to one tenth of the orig-inal value (i.e. SF = 0:1) the distances curve de-crease regularly at a slower rate. When a highershape factor is used (SF = 10:0), the algorithmconverges to a shape con�guration before the �rst10 iterations. In this case, this solution is around2.5 mm far from the base line de�ned by the ref-erence model T . This behaviour is expected, andthe reason for it is because the shape term tendsto make the model converge to the nearest shapecon�guration with respect to the dimensions in itscurrent state. Also it can be noted that when thefactor is higher the �rst iterations present abruptchanges before stabilising. These cases exemplifythe main situations found to be considered whenadjusting the shape parameter.

5.2.2 E¤ect of the Image Feature andShape Factor Variation in the Seg-mentation Results

The segmentation algorithm proposed was testedfor a given skull model using di¤erent values forthe image feature and shape factors. The valuesof the parameter combinations tested are shown intable 1 and a graph illustrating the e¤ect in thealgorithm results is presented in �gure 5.8.The graph shows the distance at each iteration j

of the resulting model Dji with respect to an aver-

age model �R. The average model �R was obtainedby applying the algorithm with SF = 1:0 and 100iterations to the deformable used in the initialisa-tion stage.

C a s e Im a g e S h a p e B e h av io u r

i Fe a t u r e s Fa c t o r o f t h e c u r v e

1 � = 1:0 = 0:0 u n sm o o t h a n d d iv e r g e n t

2 � = 1:0 = 0:1 u n sm o o t h c o n v e r g e n t

3 � = 1:0 = 1:0 sm o o t h a n d c o n v e r g e n t

4 � = 1:0 = 10:0 sm o o t h a n d c o n v e r g e n t

Table 1: Four image feature and shape factors ap-plied for segmenting a skull model

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Figure 5.8: Evaluation of results for four combina-tions of image feature and shape factors.

Case 1. In this case only image features wereused to direct the segmentation process (IFF =1:0, SF = 0:0). After initialisation, the deformablemodel evolves guided by image features de�ned byedges of the noisy skull volume and the in�uenceof its gradient vector �ow �eld [25]. Regions ofthe deformable model far from the skull volumeare pulled towards the borders of the noisy skullmodel. As a result, the surface of the model gen-erated after 20 iterations is apparently similar tothe surface of the skull volume in terms of surfacedistance but it is a "poor result" in terms of shape.The entire skull con�guration contained in the ini-tial deformable model is distorted at each step ofthe algorithm when there is no restrictions apartfrom the image features. The deformable modelevolves blindly towards the borders of the volumeof interest.

It can be observed from the graph in �gure 5.8that the algorithm converges towards a solutionnear to the �nal volume, but as the number of it-erations continue this model diverges from the so-lution. It can be observed from image in �gure5.9 (second row), that the left zygomatic bone hasan outlier due to noise components in the volume,causing the skull shape to be distorted. Also, thenasal aperture presents an unusual asymmetry dueto free deformation with no shape restrictions.

Case 2. In the segmentation process, the de-formable model is now under the in�uence of a

small amount of the shape term. With respect tocase 1, this setting modi�es the behaviour of thealgorithm by imposing a restriction in the evolu-tion of the deformable model. However, the amountof shape in�uence is small compared to the imagefeature amount (IFF = 1:0, SF = 0:1). The algo-rithm gives more priority to the image feature termresulting in a �nal con�guration that is still poorin terms of shape.Figure 5.9 (third row) shows a small improve-

ment in the �nal skull shape. This improvementcan be observed in the area of the back of the skull(contours of the parietal bone area) and the zygo-matic arch. The result is still having problems withthe noise in the left zygomatic bone area.Case 3. When a balanced combination of image

features and shape factors is used, the results areimproved in terms of global shape parameters. Fig-ure 5.9 (4th row) shows that the problem in the leftzygomatic bone has been corrected (the shape termprevents the deformable model accepting such out-liers). Also the area of the mandible is presentedwith a more regular and smooth surface. The nasalaperture maintains some symmetry and, further-more, the area of the maxilla and specially the leftsuperior teeth area are also corrected with respectto the previous con�gurations presented.Finally for the 4th case 5.9 (row 5), when there

is a high shape factor, the deformable model givespriority to the shape term, making the model con-verge to the nearer average skull shape almost in-dependently of the image features. This situationis similar to the one exposed in pure shape termvariation in the previous section.

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Figure 5.9: Results of the segmentation process combining Image Feature Factor (IFF) and the ShapeFactor (SF)

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5.2.3 Evaluation of the results in terms ofshape quality

So far, we have described the results with somedegree of "shape correctness". This concept wasanalysed in terms of visual evaluations. Observingsome general properties of a skull, such as symme-try and the proportions of some of its components,it can be assessed whether a result is correct inshape or not.In this section, a method to quantify the qual-

ity of a model in terms of its shape is proposed.To illustrate this method, we consider the resultsof three combinations of parameters exposed in theprevious section. The strategy was to calculate acommon average model as a reference. This model�R was calculated with IFF = 1:0 and SF = 1:0factors after 100 iterations from the original de-formable model used in the initialisation stage.Then, the di¤erence in position between the con-

trol points in model Ri with respect to the referencemodel �R is assessed. Also the number of iterationsneeded to stabilise the evolving deformable modelis counted.

Figure 5.10: Di¤erence in mm between the result-ing deformable model Ri and the general averagemodel �R at each iteration. The y axis representa normalised di¤erence between 0.0 and 1.0 mmcalculated with the minimum and maximum di¤er-ences found for the three models.

The graph in �gure 5.10 shows that for case 1(IFF = 1:0, SF = 0:0), the model deformed atiteration i (Ri) starts with a di¤erence around 0.5mm in distance from �R and it diverges from �R at

each step. The height of the curve and the behav-iour of its shape in terms of the di¤erence in dis-tance from R should be noted. Initially, the curvevaries abruptly, and tends to diverge thereafter.

Case 2. IFF is 1:0, and SF is 0:1. The result issimilar to case 1, although seems to stabilise at avalue of 0.9mm

For case 3, the curve is smoothly varying, andreaches a minimum at iteration 5. The distancestays below a limit of 0.2mm after 19 iterationsand continues to vary smoothly.

6 Conclusions

We have presented a new method for segmentingthe skull in MRI data. The algorithm is based ona deformable model technique. A shape term isadded to this technique to guide the segmentationprocess with statistical data. The evolution of themodel is also based on the information provided bya GVF �eld de�ned from image features. Severaltests were conducted to show the behaviour of theshape term derived from the set of training shapes.This shap term was used to control the correctnessof the �nal shape in a global way, but it can alsobe used locally to recover speci�c areas of the head.This can be acomplished by using a subset of thetraining samples corresponding to an area of inter-est.

The �nal skull models are referenced to a com-mon structure (the same topology of the triangularmesh) which is an important property in order toconduct statistical studies on these models. To-gether with the face models, the skull models willbe used for conducting and testing new craniofa-cial reconstructions approaches based on completemodels of the skull and face.

The skull models generated will be used for cre-ating a database of head models where the relationbetween the skull and face can be modelled. Fig-ure 6.1 shows a skull generated using the proposedapproach and the skin generated using marchingcubes and �gure 6.2 shows the two models super-imposed.

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Figure 6.1: An entry of the database of head mo-dles. Each head model consists of two 3D trianglemeshes representing the skull and face layers. Theskull model consists of 18,546 vertices and 36,710triangles whilst the face model consists of 25,566vertices and 49,128 triangles.

Figure 6.2: Superimposed layers of the skull andface for the �rst individual of the database.

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