Pseudo ground truth based nonrigid registration of myocardial perfusion MRI

Medical Image Analysis 15 (2011) 449–459

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Medical Image Analysis

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Pseudo ground truth based nonrigid registration of myocardial perfusion MRI

Chao Li a,⇑, Ying Sun a, Ping Chai b

a Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Republic of Singaporeb Cardiac Department, National University Heart Centre, Singapore 5 Lower Kent Ridge Road, Singapore 119074, Republic of Singapore

a r t i c l e i n f o

Article history:Received 16 December 2009Received in revised form 28 January 2011Accepted 8 February 2011Available online 16 February 2011

Keywords:Nonrigid registrationPerfusion MRIPseudo ground truthSpatiotemporal smoothness

1361-8415/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.media.2011.02.001

⇑ Corresponding author. Tel.: +65 65166332; fax: 6E-mail addresses: [email protected] (C. Li), el

[email protected] (P. Chai).

a b s t r a c t

This paper presents a novel nonrigid registration method for myocardial perfusion magnetic resonance(MR) images. To overcome the rapid intensity change due to contrast enhancement, we propose to reg-ister the observed sequence to a pseudo ground truth, which is a motion/noise free sequence that is esti-mated from the observed one, and having almost identical intensity variations as the original sequence.The pseudo ground truth and the elastic deformation fields for the observed sequence are obtained byminimizing an energy functional integrating both the registration error and the spatiotemporal con-straints on the pseudo ground truth in an expectation-maximization framework. We have tested the pro-posed nonrigid registration method on 20 cardiac perfusion MR scans. The proposed method successfullycompensated the elastic deformation of the heart in most scans according to visual validation. For quan-titative validation, we propagated manually drawn myocardial contours in one frame to other framesaccording to the deformation fields obtained by applying different registration methods. The root meansquare distance between the propagated contour and the gold standard is 2.11 mm if only global trans-lation is compensated, and 1.87 mm after nonrigid registration, as compared with 2.80 mm for serialdemons registration and 2.77 mm for a free-form deformation approach using normalized mutual infor-mation as the similarity measure, both of which adversely increased the error due to misregistration.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Dynamic contrast enhanced (DCE) magnetic resonance imaging(MRI) has shown promising results for detecting abnormalities inmyocardial blood flow, and demonstrated great potential for diag-nosing cardiovascular diseases (Al-Saadi et al., 2000; Bertschingeret al., 2001; Klassen et al., 2006). Although cardiac perfusion MRimages are acquired using an electrocardiogram gating techniquein order that each slice is always scanned during the same phaseof the cardiac cycle, patient breathing during image acquisition of-ten causes large variations of the heart position in different frames.In order to access the perfusion signal of the myocardium, rigidregistration is necessary to compensate this breathing motion(Breeuwer et al., 2001; Gupta et al., 2003; Buonaccorsi et al.,2005; Adluru et al., 2006; Milles et al., 2008). Gao et al. (2002)demonstrated that respiration not only changes heart positionbut also induces cardiac deformation. Besides, arrhythmia alsoleads to heart shape change in perfusion sequences. Due to thesenonrigid heart deformations, compensating only for global motionis insufficient to accurately retrieve perfusion signals around myo-cardial boundary regions such as the subendocardium, where

ll rights reserved.

5 [email protected] (Y. Sun),

ischemia first appears (Reimer et al., 1977). Therefore, nonrigidregistration of time-series images is important for myocardialischemia diagnosis (Yang et al., 1998; Gao et al., 2002; Ólafsdóttiret al., 2006; Wollny et al., 2008; Hennemuth et al., 2008).

The challenge in perfusion image registration mainly arisesfrom rapid intensity changes of heart ventricles during the wash-in/wash-out of the contrast agent. Normalized mutual information(NMI) is often used in registering images with different intensitydistributions (Maes et al., 1997; Rueckert et al., 1999), but it hasa well-known drawback of being computationally expensive. ForDCE breast MR images with spatially varying enhancement, Zhenget al. (2007) developed a two-step registration algorithm that iter-atively recovers the enhancement map and registers the de-enhanced images. Ebrahimi and Martel (2009) recently proposeda partial differential equation framework that simultaneously esti-mates the enhancement map and the displacement vector field. InZheng et al. (2007) and Ebrahimi and Martel (2009), validation isonly performed on pairs of selected images rather than an entiresequence.

To minimize the intensity difference between the reference andfloating images, one approach to registration of a dynamic se-quence is to successively register every two consecutive frames(e.g., Yang et al., 1998; Breeuwer et al., 2001; Gupta et al., 2003;Hennemuth et al., 2008). This serial registration scheme howevertends to accumulate registration errors. Wollny et al. (2008)

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450 C. Li et al. / Medical Image Analysis 15 (2011) 449–459

proposed to select several key frames according to the breathingperiodicity as the reference frames, but this method still requiresregistration of images from different perfusion phases.

Several pieces of work address this intensity variation problemby using an enhancement-driven synthetic sequence to specify thetarget image for all the frames. In Buonaccorsi et al. (2005) andAdluru et al. (2006), the intensity-time curve at each pixel is fittedto a dynamic contrast-enhanced MRI model (Tofts et al., 1999).Accordingly, the observed sequence is rigidly registered to theresultant synthetic series which is assumed to be free of motion.These model-based registration methods have the limitations thatthe fitting is complex and an input function of the contrast agentconcentration in the arterial supply is required. Stegmann andLarsson (2003) proposed to register perfusion sequence by usingCluster-aware Active Appearance Model which is built from anannotated training set. Milles et al. (2008) used Independent Com-ponent Analysis to compensate rigid heart motion in perfusionscans, whereas Melbourne et al. (2007) used Principal ComponentAnalysis to compensate liver deformation in DCE-MRI.

We herein propose a novel registration method that overcomesthe challenges arising from time-varying intensity by using pseudoground truth. The pseudo ground truth is an estimate of the imagesequence that would have been acquired without being affected bymotion or noise during acquisition. We design an energy functionalthat integrates both nonrigid registration and pseudo ground truthestimation, which can be minimized iteratively by solving a systemof linear equations and applying nonrigid registration to corre-sponding pairs of images between the observed sequence andthe pseudo ground truth sequence (see Fig. 1). In contrast to pairsof images within the observed sequence, these corresponding pairsof images have similar intensities, and therefore the registrationproblem is greatly eased. Some preliminary results of this methodhave been presented in Li and Sun (2009). Here, we extend our pre-vious work by: (a) incorporating the information from heart ventri-cle segmentation in the design of the energy functional; (b)applying conjugate gradient method (Hestenes and Stiefel, 1952)to accelerate the pseudo ground truth estimation; and (c) usingan edge-emphasized demons registration algorithm to preservetexture information.

The remaining of the paper is organized as follows. Section 2 de-scribes our nonrigid registration algorithm. Section 3 presents theexperimental results on both artificial and real cardiac perfusion

(1) (2)

Fig. 1. Algorithm outline: (a) the sequence after rigid registration; (b) the pseudo groundregistration until convergence.

MRI scans, followed by discussion in Section 4 and conclusion inSection 5.

2. Method

Let the static frame refer to a user selected frame to which allthe other frames will be registered. Our registration frameworkconsists of three steps: (a) initial alignment by identifying the glo-bal translation of the ROI in each frame with respect to the staticframe; (b) semi-automatic segmentation of the left ventricle (LV),the right ventricle (RV), and the LV myocardium; and (c) estima-tion of local elastic deformation by minimizing an energy func-tional that incorporates errors resulting from both registrationand pseudo ground truth fitting.

2.1. Initial alignment

The purpose of this step is to bring the registration parameterscloser to the global optimal values, and thus making our registra-tion scheme more effective and efficient (Moelich and Chan,2003). Given a rectangular ROI in the static frame, initial alignmentseeks to align the ROI across the MR image sequence. In perfusionMRI, the intensities of the acquired time-series images change rap-idly as the contrast agent flows through the heart. However, theorientation of the edges along tissue boundaries tends to remainconsistent across the image sequence provided that the heart doesnot undergo significant elastic deformation. Based on this observa-tion, we apply the algorithm proposed in Sun et al. (2004a) to trackthe ROI with integer pixel shifts using a gradient-based similaritymeasure.

Let hs(x,y) and Ms(x,y) respectively denote the direction and themagnitude of the spatial intensity gradient at pixel (x,y) in the sta-tic image, likewise hc(x,y) and Mc(x,y) for the current frame to becalibrated with the static frame. We obtain the translation (dx,dy)by maximizing the similarity metric:

Sðdx; dyÞ ¼Xðx;yÞ2R

xðx; y; dx;dyÞ cosð2Dhðx; y; dx; dyÞÞ; ð1Þ

where R represents the ROI, x(x,y;dx,dy) is a weight function andDh(x,y;dx,dy) is the angle difference:

(a)

(b)

truth sequence. We repeat (1) pseudo ground truth fitting and (2) corresponding image

C. Li et al. / Medical Image Analysis 15 (2011) 449–459 451

xðx; y; dx;dyÞ ¼ Mcðxþ dx; yþ dyÞMsðx; yÞPðx;yÞ2R

Mcðxþ dx; yþ dyÞMsðx; yÞ; ð2Þ

Dhðx; y; dx; dyÞ ¼ hcðxþ dx; yþ dyÞ � hsðx; yÞ: ð3Þ

Fig. 2 displays the results after initial alignment for five selectedframes from a myocardial perfusion MRI scan, in which the boundingbox of the ROI has been shifted to the best match location in eachframe. By aligning the ROI from different frames, we obtain a roughlyregistered ROI sequence that still contains local elastic deformation.

2.2. Heart ventricle segmentation

Segmentation information is important in defining the spatio-temporal smoothness constraints because the spatial smoothnessconstraint should be restricted within the same tissue to avoidblurred boundaries in the pseudo ground truth, while the temporalsmoothness constraint should be adaptive to different tissue typesto account for their distinct dynamic models.

We use a multistage level set method to sequentially segmentthe LV, the RV, and the myocardium. The segmentation is semi-automatic and requires two clicks of user inputs: one inside theLV and the other inside the RV. The energy functional is definedaccording to the intensities of the static image and the temporaldynamics of the entire sequence. Let C represent the active con-tour, i.e., the zero level set. Let X1 and X2 respectively denotethe region inside and outside C, and u(x,y) the intensity valueof pixel (x,y) in the static frame. We define the energy functionalas:

JðCÞ ¼ k1 � LengthðCÞ þ k2 � AreaðX1Þ

þXi¼1;2

ZXi

�k3 ln pðuðx; yÞjXiÞ þ k4disðvðx; yÞ; �viÞ½ �dxdy; ð4Þ

where p(u(x,y)jXi) is the Gaussian intensity model ofXi;disðvðx; yÞ; �viÞ is the distance between the intensity-time curveof (x,y) and the average intensity-time curve in Xi (Sun et al.,2004b), and kk (k = 1,2,3,4) are term weights. In (4), the first twoterms are regularization terms controlling the smoothness of thecontour (Chan and Vese, 2001); the other two terms are region-based terms that segment the image into homogeneous regionswith respect to the intensities of the static image and the temporaldynamics. For epicardium segmentation, we employ the samemethod as described in Li et al. (2009), which overcomes complexbackground and low contrast by using a dual-background intensitymodel and the myocardium thickness constraint.

g¼vecðgÞ¼ ½gð1;1;1Þ gð2;1;1Þ . . . gðNi;1;1Þ . . . gð1;Nj;1Þ gð2;Nj;1Þ . . . gðNi;Nj;1Þ . . . gð1;1;NtÞ . . . gðNi;1;NtÞ . . . gðNi;Nj;NtÞ�T; ð6Þ

Fig. 3 shows our segmentation results which are fairly accuratefor most data sets. Since our goal here is to explore how myocardialsegmentation benefits the pseudo ground truth based registration,we use a relatively simple segmentation approach. Improving

Fig. 2. Initial alignment of a myocardi

segmentation, for instance, by incorporating shape prior or train-ing, is likely to further enhance the registration performance.

2.3. Nonrigid registration

As aforementioned in Section 1, we use a pseudo ground truthsequence that is free of motion/deformation to guide the nonrigidregistration of the perfusion sequence. We estimate the pseudoground truth sequence from the observed sequence by imposingspatiotemporal smoothness constraints, which are distinctiveattributes that differentiate between sequences with and withoutmotion/deformation. In the spatial domain, homogenous regionsin the static frame should remain homogenous in all the otherframes for a motionless sequence, which however does not applyto sequences with motion/deformation. In the temporal domain,signals in a motionless sequence are smooth as the contrast agentgradually perfuses through the heart whereas in a sequence withmotion/deformation, signals for pixels around myocardial bound-aries may capture intensities of different tissues and are thereforeoscillating as illustrated in Fig. 4.

Once we estimate a motion-free pseudo ground truth using spa-tiotemporal smoothness constraints, we register each image in theobserved sequence to its counterpart in the pseudo ground truthimage sequence. This bypasses the problem of having unknownintensity variations between different frames, because under ourscheme registration is between images with almost identicalintensity distributions.

We formulate the problem as follows. Given an observed imagesequence g, we solve for a pseudo ground truth sequence f and anonrigid deformation function H that minimize the following en-ergy functional:

EðHðgÞ; f Þ ¼ EdðHðgÞ; f Þ þ Esðf Þ þ Etðf Þ; ð5Þ

subject to the constraint on H that its underlying displacement field iszero for the static frame, and is smooth for the remaining frames. In(5), Ed is the data fidelity term measuring the difference betweenthe pseudo ground truth sequence f and the nonrigidly deformed se-quence H(g); Es is the spatial smoothness constraint penalizing theintensity difference between neighboring pixels of the same tissuetype; and Et is the temporal smoothness constraint penalizing thefirst and second order derivatives of the perfusion signal of each pixel.

To rewrite the energy functional in a matrix-vector form, eachimage sequence is represented as a column vector, e.g., if g(i, j, t)is the intensity at pixel (i, j) in MRI frame t, then the column vectorg is given by (6):

where Ni and Nj are respectively the number of rows and col-umns of each image, and Nt is the number of frames. Similarly,we define f = vec(f) as the column vector for the pseudo groundtruth f.

al perfusion MR image sequence.

Fig. 3. Results of heart ventricle segmentation.


2.3.1. Data fidelity termLet ~g ¼ HðgÞ denote the image sequence obtained by deforming

g with the deformation field H. The data fidelity term Ed is the sum-squared intensity difference between ~g and f:

EdðHðgÞ; f Þ ¼ Edð~g; fÞ ¼ ð~g� fÞTð~g� fÞ; ð7Þ

where ~g ¼ vecð~gÞ ¼ vecðHðgÞÞ.

2.3.2. Spatial smoothness constraintThe spatial smoothness constraint is used to address the ambigu-

ity in estimating the perfusion signals around the boundaries. Con-sider the intensity-time curve of a pixel around the boundarybetween the LV cavity and the myocardium as an example: due tothe motion and/or deformation of the heart, the intensity-time curveat the boundary oscillates between the perfusion models of the LVand the myocardium (see the blue circles in Fig. 4). Buonaccorsiet al. (2005) and Adluru et al. (2006) locally (pixel-by-pixel) fit theintensity-time curves to pharmacokinetic models, which suffersfrom the ambiguity of oscillating intensity-time curves around theboundary and most likely results in a compromise between thetwo models. Since the intensity of this compromised signal is in be-tween the intensities of the two neighboring models, these local

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frame Number

Nor

mal

ized

Inte

nsity

RVLVMyocardiumBoundary

Fig. 4. Intensity-time curves of pixels located inside the RV, LV, myocardium, and atthe boundary between the LV and the myocardium. In our registration framework,we normalize the intensities into the interval of [0,1].

fitting approaches generate a synthetic sequence with blurredboundaries in the presence of motion, which will potentially leadto misregistration, especially for nonrigid registration that mainlyrelies on local information.

Instead of estimating perfusion signals pixel-by-pixel, we con-sider neighboring curves simultaneously in estimating perfusionsignals by imposing the spatial smoothness constraint on the pseu-do ground truth sequence, such that perfusion signals of non-boundary pixels, which are less likely affected by motion, help toresolve the aforementioned ambiguity. Specifically, we penalizethe sum of weighted intensity differences between each pixeland its nearest neighbors:

Esðf Þ ¼ EsðfÞ ¼XK

k¼1

ðDskfÞ TWkðDs

kfÞ; ð8Þ

where K is the number of neighboring pixels being considered, andin this work we use the 4-neighborhood system, i.e., K = 4; Ds

k is thefirst order spatial derivative operator along the direction betweeneach pixel and its kth neighbor, and Wk is the corresponding weightmatrix to ensure that the intensity difference between pixels fromdifferent tissue types are not penalized.

Assuming that pixels of the same tissue type have similar inten-sities in the static frame and their perfusion signals exhibit similartemporal dynamics, we define the spatial weight matrix as:Wk = diag(vec(wk)) and

wkði;j;tÞ¼aqijk exp � gði;j;tsÞ�gðik ;jk ;tsÞ½ �2

2r2

n o; if qijk>qth & Lij¼Likjk

0; otherwise

(;

ð9Þ

where a is a positive scalar controlling the term weight, qijk denotesthe correlation coefficient between the intensity-time curves of pix-el (i, j) and its kth neighbor (ik, jk); ts is the static frame number; Llabels the RV, the LV, and the myocardium, which is obtained bya multistage level set method described in Section 2.2; and qth

and r2 respectively define the minimal signal similarity and theintensity variance within the same tissue type. In our implementa-tion we set qth = 0.85 and r = 0.05.


2.3.3. Temporal smoothness constraintThe temporal smoothness constraint essentially uses the tem-

poral neighborhood of each frame to estimate its counterpart inthe pseudo ground truth. Since the contrast agent has no effecton the background and in the pre-contrast phase, the perfusion sig-nals in the background and during pre-contrast periods should beapproximately constant, i.e., their first order temporal derivativesshould be close to zero. Moreover, as the contrast agent graduallyperfuses through the heart, the perfusion signals should be approx-imately piece-wise linear, and thus the second order temporalderivative of the pseudo ground truth should be penalized. Thetemporal smoothness constraint is therefore defined as:

Etðf Þ ¼ EtðfÞ ¼ ðDt1fÞTWtðDt

1fÞ þ b2ðDt2fÞTðDt

2fÞ; ð10Þ

where Dt1 and Dt

2 are the first and the second order temporal deriv-ative operators, Wt = diag(vec(wt)), and

wtði; j; tÞ ¼b1; if ði; j; tÞ 2 pre-contrast phaseb1; if Lij ¼ background0; otherwise

8><>: : ð11Þ

For each tissue type, we use its average intensity-time curve toidentify the pre-contrast phase.

Similar as a in (9), positive scalars b1 in (11) and b2 in (10) con-trol the corresponding term weights.

2.3.4. Energy minimizationIn our implementation, the energy functional defined in (5) is

minimized by iteratively solving for the optimal H and f in anexpectation-maximization fashion.

In each iteration, first we keep H fixed (hence obtaining a de-formed sequence ~g), and estimate the pseudo ground truth f̂ byminimizing

E ¼ ð~g� fÞTð~g� fÞ þXK

k¼1

ðDskfÞTWkðDs

kfÞ þ ðDt1fÞTWtðDt

1fÞ

þ b2ðDt2fÞTðDt

2fÞ: ð12Þ

This minimization requires solving a system of linear equations:

IþXK

k¼1

Dsk

TWkDsk

� �þ Dt

1TWtD

t1 þ b2Dt

2TDt

2

" #f̂ ¼ ~g: ð13Þ

Let A ¼ IþPK

k¼1ðDsk

TWkDskÞ þ Dt

1TWtD

t1 þ b2Dt

2TDt

2, Eq. (13) becomesAf̂ ¼ ~g. In our previous work (Li and Sun, 2009), the linear systemAf̂ ¼ ~g was solved by Gaussian elimination, which is time consum-ing due to the large order of A. By further analysis of A, we find itsymmetric and positive definite. Therefore, in this work, we usethe conjugate gradient method, which is much more efficient thanGaussian elimination, to solve the linear system in (13).

Next, we calibrate the pseudo ground sequence and the ob-served sequence by registering the static image in f̂ to its counter-part image in g and then applying the resulting deformation fieldto all the images in f̂ , to ensure that the underlying deformationfield for the static frame is zero.

Finally, we keep f̂ fixed, and register each image in g to its coun-terpart in the pseudo ground truth sequence f̂. The resultant defor-mation fields completely and uniquely define the deformationfunction H, and hence ~g can be updated accordingly.

Here we adopt the demons algorithm described in Wang et al.(2005) to register corresponding images. Since the observed se-quence contains random noise whereas the pseudo ground truthis spatiotemporally smooth, the corresponding images in the pseu-do ground truth and the observed sequence differ slightly intexture. These texture differences may generate small optical flowin homogeneous regions, and hence lead to texture distortion

in the registered image. To preserve image texture, we regularizethe transformation field by using a weighting map derived fromthe gradient magnitude of the observed image, such that it empha-sizes large optical flows around strong edges while suppressingsmall optical flows in homogenous regions. Thus, the adapted de-mons algorithm is called edge-emphasized demons algorithm.

The initial condition is ~g ¼ g, i.e., the original observed imagesequence. The matrices Wt and Wk(k = 1, . . . ,K) are re-estimatedusing the updated ~g at each iteration. To obtain the optimal defor-mation, the evolution should continue until E cannot be further re-duced. In our implementation, we empirically fix the iterationnumber to 3, which generates adequately accurate registration re-sults and saves computation time. The energy minimization algo-rithm is summarized in Algorithm 1.

Algorithm 1. Energy minimization

Require: H0

1: H = H0

2: while not converged do3: ~g ¼ HðgÞ4: Update Wt,Wk

5: A ¼ IþPK

k¼1ðDsk

TWkDskÞ þ Dt

1TWtDt

1 þ b2Dt2

TDt

2

6: Solve Af̂ ¼ ~g for f̂

7: Register f̂ static to gstatic and obtain hstatica

8: Use hstatic to calibrate f̂

9: Register g to the calibrated f̂ and obtain H10: end while

af̂ static represents the static frame in f̂ , likewise gstatic and hstatic.

3. Experimental results

The proposed algorithm has been implemented in MATLAB (TheMathWorks, Natick, MA, USA) interfacing some external C libraries,and has been tested with both artificial and real perfusion scans.

3.1. Artificial perfusion scans

In order to validate whether local perfusion information is lostduring our nonrigid registration, we first apply our method to non-rigidly register an artificial sequence with known perfusion signalsand deformations. The artificial perfusion sequence is generated asfollows. We first extract perfusion signals for LV, RV, myocardium,and the ischemic tissue from a real perfusion scan by averagingsignals over a number of corresponding pixels that are not affectedby motion. Then we manually drew the mask for each region andgenerate an artificial motionless perfusion scan using the extractedsignals plus Gaussian noise. We nonrigidly deform all the framesexcept the static frame by using B-spline free-form deformations(FFD) (Rueckert et al., 1999) with random mesh pointdisplacement.

Fig. 5a–c respectively show a typical frame of the input artificialperfusion sequences with deformations, without deformations,and the corresponding deformation field. The frame is chosen forthe one in which the ischemic tissue underwent relatively largedisplacement.

Fig. 5d–f respectively show the corresponding frame of the reg-istered perfusion sequence, the pseudo ground truth sequence andthe recovered deformation field. Because the spatial smoothnessconstraint cannot differentiate ischemic region from the myocar-dium as their intensity-time signals closely correlate to each other,

(c)(b)(a)

(f)(e)(d)Fig. 5. Results for artificial data. (a)–(f) respectively show the 34th frame of the artificial sequence, the artificial motionless sequence, the input deformation field, thenonrigidly registered sequence, the estimated pseudo ground truth and the recovered deformation field.

0 10 20 30 40 50 600

0.1

0.2

0.3

Frame Number

Nor

mal

ized

Inte

nsity

Sequence without DeformationSequence with DeformationAfter Non−rigid Registration

Fig. 6. Average perfusion signals in the ischemic tissue for the artificial experiment.


the boundary of the ischemic tissue (the dark rectangular region atthe right hand side of the myocardium) is slightly blurred. Never-theless, the registered image has fairly good sharpness because ourdemons registration method relies more on strong edges (i.e., myo-cardial boundaries). The average perfusion signals within theischemia for the artificial motionless sequence, nonrigidly de-formed sequence, and the output of our method are shown inFig. 6. As shown, nonrigid deformation induced signal change inthe ischemic region, whereas our nonrigid registration methodsuccessfully recovered the signal,1 i.e., the local perfusion informa-tion is maintained notwithstanding the spatial smoothnessconstraint.

3.2. Real perfusion scans

The real perfusion scans comprise of 20 slices of cardiac perfu-sion images from 13 patients, acquired by Siemens Sonata/AvantoMR scanners following bolus injection of Gd-DTPA contrast agent.

1 The slight intensity mismatch between the registered sequence and the artificialmotionless sequence is due to the partial volume effect.

The matrix size ranges from 128 � 88 to 320 � 260 with pixelspacing ranging from 1.04 � 1.04 to 2.90 � 2.90 mm2. Each perfu-sion scan contains 33–60 frames.

For a 60-frame scan with an ROI size of 60 � 60, the computa-tion time ranges from 2 to 3 min on a desktop PC (intel Core 2Duo 3.0 GHz, 4 GB DDR2 RAM, codes were not designed to takeadvantage of multicore processor system). Specifically, globaltranslation is completed in 2 s, heart ventricle segmentation takesabout 4 s, the pseudo ground truth fitting takes less than 1 secondfor each iteration, and one iteration of nonrigid registration re-quires a running time of 18–67 s, depending on the magnitude ofdisplacement. Comparing the computation time for each step, wefind that the overall running time of our method highly dependson the nonrigid registration algorithm it uses and the iterationnumber, while the pseudo ground truth fitting incurs little extracomputation. In contrast to using the conjugate gradient method,fitting the pseudo ground truth by Gaussian elimination as in ourprevious work (Li and Sun, 2009) takes about 30 s per iteration,which almost doubles the total computation time. Moreover,Gaussian elimination requires more than 1 GB memory to solvefor the pseudo ground truth, while the conjugate gradient methodonly needs about 10 MB and is hence a great improvement in prac-tical value.

For all the data sets in our study, large-scale translation can beidentified reliably using the global template matching method de-scribed in Sun et al. (2004a) and the elastic deformation is mostlycompensated using our method according to visual validation fromvideos. Next, we present the experimental results in two aspects:contour propagation and comparison of perfusion signals.

3.2.1. Contour propagationSince we calibrate the pseudo ground truth and the rigidly

translated sequence w.r.t. the static frame, the transform field be-tween the two sequences represents the transformation betweenthe static frame and any other frames, i.e., the inter-frame


deformations. Given the manually drawn boundaries of the myo-cardium in the static frame as landmarks, one can propagate thesecontours to other frames according to the deformation fields ob-tained by our registration method. Therefore, one way to evaluatethe performance of the nonrigid registration algorithm is to verifywhether the propagated contours well delineate the myocardialboundaries in other frames.

The first two rows of Fig. 7 compare the propagated contours be-fore and after applying nonrigid registration for four frames, com-prising of one pre-contrast frame and three post-contrast frames,from a real cardiac MR perfusion scan. As shown in the top row,the contours before nonrigid registration do not lie exactly at theboundaries of the LV myocardium. In contrast, the contours in thesecond row, after nonrigid registration using the proposed method,delineate well both endocardial and epicardial boundaries.

To further demonstrate the performance of the proposed meth-od, we compare our results with those obtained by serial registra-tion using the edge-emphasized demons algorithm aforementioned in Section 2.3.4, in which every two consecutive framesare registered to propagate from the static frame to the rest of thesequence (this approach will be referred to as serial demons fordescription convenience). As shown in the third row of Fig. 7, byusing serial demons, the propagated contours in the 21st, 36th,

9

9

9

9

18

18

18

18

Fig. 7. Contour propagation for one pre-contrast frame and three post-contrast framesregistration (top row), contours propagated by our method (second row), serial demons

and 40th frames are away from the true boundaries in the regionsindicated by arrows due to the accumulation of registration errors.We have also applied the FFD registration method (Rueckert et al.,1999) using NMI as the similarity measure, in which the static frameis used to register all the other frames in the sequence (this approachwill be referred to as NMI-based FFD for description convenience). Asshown in the bottom row of Fig. 7, the propagated myocardial con-tours do not delineate the boundaries as accurately as those ob-tained by our method (see the second row of Fig. 7). Fig. 9 showsthe propagated contours for another four cardiac perfusion scans,which qualitatively sustains the effectiveness of our nonrigid regis-tration method.

Fig. 8 compares a representative frame of the pseudo groundtruth and the propagated contours generated by the methods pre-sented in this paper and our previous work (Li and Sun, 2009). Asshown in the left column, without using the segmentation infor-mation, the pseudo ground truth obtained by Li and Sun (2009)is not as sharp as that obtained by our current approach (e.g.,around the top epicardial region). Consequently, our current meth-od produces better epicardial contour propagation result as shownin the right column.

For quantitative evaluation, a cardiologist manually drew myo-cardial contours for the slices in which the endocardium and/or

24

24

24

24

47

47

47

47

from a real patient cardiac MR perfusion scan: contours before applying nonrigidregistration (third row), and NMI-based registration (bottom row).

Fig. 8. Comparison of the estimated pseudo ground truth sequences (left) and thepropagated contours (right) obtained using the methods presented in Li and Sun,2009 (top), and this paper (bottom).

(a) (b)Fig. 9. Contour propagation for four cardiac scans using our method. Column (a) shows t(b)–(d) show the propagated myocardial contours before, during, and after the first pas


epicardium are visible, and then measured the root mean square(RMS) distance from the manually drawn contours to the propa-gated contours similar as in Jolly et al., 2009. As shown inTable 1, our method improves the accuracy of the propagated con-tours for both the endocardial and epicardial boundaries and out-performs serial demons and NMI-based FFD approaches.Compared to the contour propagation using global translationonly, our nonrigid registration method decreased the RMS distancefrom 1.18 pixels (2.11 mm) to 1.04 pixels (1.87 mm), whereas theserial demons and NMI-based FFD methods increased the RMS dis-tance to 1.57 pixels (2.80 mm) and 1.48 pixels (2.77 mm) due tomisregistration. The inverted cumulative histograms for the fourmethods and no registration are plotted in Fig. 10. As illustratedby Jolly et al. (2009), a point (x,y) on the curve indicates that x%of all distances are not greater than y pixels, meaning that the bot-tom-right curve corresponds to the propagation method providingthe best match to the manually drawn contours. As shown inFig. 10, our registration method generates the best propagation re-sults: 74.39% of the distances are no greater than 1 pixel, as com-pared with 66.08%, 58.06%, and 53.87%, respectively for globaltranslation, NMI-based FFD, and serial demons. The RMS distancesfor respective data sets are shown in Table 2. For 18 out of 20 datasets, contours propagated by our method better match the groundtruth than only using global translation. Additionally, our method

(d)(c)he static frame, on which the myocardial boundaries are manually drawn; columnss of bolus.

Table 1The RMS distances (pixels/mm) between the manually drawn contours and the propagated contours for the endocardium, epicardium, and all the contours. The distances aremeasured in terms of pixels and millimeters (mm) separately.

Contour Without registration Global translation Proposed method Li and Sun (2009) Serial demons NMI-based FFD

Endo- 2.57/3.80 1.12/2.00 0.93/1.70 0.99/1.81 1.49/2.56 1.34/2.53Epi- 2.51/4.03 1.22/2.18 1.11/1.97 1.13/2.04 1.62/2.95 1.57/2.91Overall 2.54/3.94 1.18/2.11 1.04/1.87 1.08/1.95 1.57/2.80 1.48/2.77

0 10 20 30 40 50 60 70 80 90 1000

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ixel

s]

Proposed MethodLi & Sun (2009)NMI−based FFDSerial DemonsGlobal TranslationWithout Registration

Fig. 10. The inverted cumulative histogram for distances between the propagatedcontours and the manually drawn contours. Circles highlight the proportions ofdistances that are not greater than 1 pixel.

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(b)(a)Fig. 11. Comparison of intensity-time curves before and after nonrigid registration.(a) the static frame of one perfusion sequence, and (b) comparison of the intensity-time curves for the myocardial pixel marked in (a).


significantly surpassed serial Demons and MI-based FFD ap-proaches for most data sets. Table 2 also shows the improvementover our previous approach (Li and Sun, 2009), especially for data-sets #7 and #8. In datasets #6 and #11, distances between thepropagated contours and manually drawn ones are slightly in-creased after nonrigid registration. This is mainly due to the incon-sistency of the manually drawn contours between the referenceframe and other frames. In both datasets, there was hardly any vis-ible epicardial fat over the lateral wall, so the epicardial border def-inition over the lateral wall becomes indistinct as contrast washedout of the myocardium, thus making it difficult to keep manuallydrawn myocardial boundaries consistent. Besides, the myocardialboundaries in dataset #6 are very blurry, which poses further dif-ficulty for both ‘ground truth’ drawing and our nonrigid registra-tion algorithm.

3.2.2. Comparison of intensity-time curvesFig. 11 shows the intensity-time curves before and after non-

rigid registration for a pixel near the myocardial boundary. As

Table 2Comparisons of the RMS (pixels/mm) distances for respective data sets. Best results for re

Data Without registration Global translation Proposed met

#1 3.76/7.06 1.26/2.35 1.09/2.04#2 1.19/3.26 0.92/2.51 0.73/1.99#3 0.93/1.74 0.90/1.68 0.84/1.58#4 0.95/2.75 1.00/2.89 0.87/2.52#5 1.36/2.02 1.27/1.89 1.13/1.68#6 2.06/3.23 1.41/2.20 1.47/2.29#7 2.83/3.98 0.87/1.23 0.66/0.93#8 1.62/3.04 1.63/3.05 1.25/2.34#9 1.26/2.37 1.06/1.98 0.91/1.70#10 1.29/3.52 0.79/2.16 0.73/2.00#11 1.56/2.92 1.37/2.56 1.41/2.64#12 1.43/2.69 1.38/2.59 1.28/2.41#13 1.77/1.85 1.29/1.35 1.03/1.07#14 1.22/1.71 0.67/0.94 0.57/0.80#15 2.27/6.20 1.09/2.97 1.00/2.73#16 2.27/5.38 1.22/2.90 1.02/2.43#17 3.91/4.65 1.13/1.34 0.95/1.13#18 4.58/5.44 1.44/1.71 1.08/1.28#19 1.54/3.86 1.05/2.62 0.93/2.32#20 3.99/4.74 1.35/1.60 1.21/1.44

shown in Fig. 11b, the perfusion signal after global translation stillexhibits significant oscillations due to local deformation. In con-trast, after compensating for the local deformation by performingnonrigid registration, the intensity-time curve becomes smootherat frames where the LV undergoes noticeable local deformation.Note that the remaining small local oscillations are caused by im-age noise.

In addition to the intensity-time curve, we also compare theperfusion parameter, i.e., the normalized upslope (Milles et al.,2008), in the subendocardial region before and after nonrigid reg-istration. As shown in Fig. 12, for most data sets, the standard devi-ation of the upslope is greatly reduced, which indicates that theperfusion parameter in the subendocardial region is less noisy afternonrigid registration. We believe such reduction of the standarddeviation is due to successful compensation of nonrigid deforma-tion rather than erroneous registration, because the nonrigidly reg-istered sequences are obtained by pure intra-frame warpingwithout temporal smoothing. Moreover, no texture distortion ofthe myocardium is observed in registered sequences, which havebeen validated by a cardiologist. Fig. 12 also shows that the

spective data sets are highlighted in bold.

hod Li and Sun (2009) Serial demons NMI-based FFD

1.15/2.15 1.21/2.26 1.33/2.500.73/1.99 1.17/3.20 1.25/3.430.85/1.60 0.88/1.65 1.16/2.180.93/2.70 0.98/2.83 0.99/2.871.13/1.68 1.57/2.33 1.19/1.771.41/2.20 1.87/2.92 1.61/2.520.77/1.08 1.48/2.08 1.14/1.601.61/3.02 2.68/5.03 1.80/3.370.98/1.84 1.52/2.85 2.91/5.450.79/2.16 0.90/2.45 1.21/3.301.37/2.57 1.96/3.67 2.86/5.361.23/2.30 1.79/3.35 1.61/3.031.06/1.10 1.48/1.54 1.94/2.030.65/0.91 0.85/1.20 1.07/1.511.05/2.88 1.84/5.04 1.56/4.271.07/2.54 0.92/2.18 1.05/2.490.94/1.12 1.75/2.08 1.26/1.501.14/1.36 1.63/1.93 0.96/1.140.96/2.40 1.16/2.90 1.28/3.191.17/1.39 1.50/1.78 1.17/1.39

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Fig. 12. Comparison of the statistics of the normalized upslope in subendocardialregion before and after nonrigid registration.


average normalized upslope before nonrigid registration is greaterthan that after nonrigid registration. This is because without com-pensating elastic deformations, myocardial perfusion signals arecontaminated by LV signals, which have much greater upslopesthan normal myocardial signals.

4. Discussion

The proposed nonrigid registration method uses spatiotemporalsmoothness constraints to compensate for the elastic deformationof perfusion sequences by iterative optimization. In contrast to tra-ditional registration approaches that obtain the deformation fieldby pairwise registration of the observed images, our method seeksthe global optimal deformation for the entire sequence by intro-ducing the pseudo ground truth. As the intensity variations ofthe pseudo ground truth and the observed sequence are almostidentical, it becomes not necessary to use multi-modality registra-tion algorithms (such as NMI-based methods) that are computa-tionally expensive in general. Consequently, it enables us toapply existing intensity based registration algorithms that aremore computationally efficient.

The temporal smoothness constraint essentially uses the tem-poral neighborhood of a frame to estimate its counterpart in thepseudo ground truth. Note that the contrast variations for pre-and post-bolus arrival frames are not linear, which indicates thatthe perfusion signal is piece-wise linear. Accordingly, there are min-or intensity differences between the pseudo ground truth and theobserved signals for those frames due to smoothing. Such scenariois similar to the regularization in the registration approach wherethe first order or second order derivative penalty is commonly usedalthough the optimal deformation is neither constant nor linear forall the pixels. In fact, the pseudo ground truth here is only used tofacilitate nonrigid registration instead of perfusion signal extrac-tion. We assume that most registration methods can tolerate suchminor intensity differences, and our experimental results revealthat the demons algorithm satisfies this assumption.

By incorporating the spatial smoothness constraint, the pseudoground truth fitting for a pixel uses the major signal of the regionto resolve the ambiguity caused by deformation (Section 2.3.2). Thismay lead to slight texture difference between the pseudo groundtruth sequence and the observed sequence, and we use an edge-emphasized demons method to overcome this texture mismatch.In order to maintain the sharpness of strong edges, like myocar-dium boundaries, we incorporate heart ventricle segmentation toensure that pixels from different regions are not smoothed at all.

Compared with previous research using synthetic sequence tofacilitate registration in Buonaccorsi et al. (2005), Adluru et al.

(2006), Melbourne et al. (2007) and Milles et al. (2008), the advan-tage of our method is threefold:

(a) Our registration method utilizes spatiotemporal smoothnessconstraints in generating the synthetic sequence, i.e., thepseudo ground truth. It is more generalized than pharmaco-kinetic model based approaches (Buonaccorsi et al., 2005;Adluru et al., 2006).

(b) Instead of analyzing the intensity-time curves indepen-dently, we introduce the spatial smoothness constraint inpseudo ground truth fitting, so that the estimated signalfor a pixel depends on not only its own intensity-time curve,but also its neighbors’. This helps avoid blurred boundariesin the synthetic sequence, and consequently prevents themethod from incorrectly converging to local optima.

(c) The method has proven to be capable of compensating fornonrigid deformation which is common in cardiac perfusionstudies, while most of the existing synthetic sequence basedmethods focus only on rigid registration. Although Mel-bourne et al. (2007) address nonrigid motion, their methodrequires that there is no periodic motion present in thesequence, despite that periodic motion is quite common inperfusion studies due to patient breathing. Our method doesnot have this limitation.

One limitation of the proposed method lies in the increasedcomputational complexity associated with using iterative optimi-zation. Introducing the pseudo ground truth overcomes the inten-sity variation problem, however, the energy functional of nonrigidregistration is not minimized in one attempt, but in an iterativecoarse-to-fine manner. Although empirically three iterations givesatisfactory results, the absolute convergence of the algorithmmay require more iterations.

An alternative solution circumventing iterative optimization isto directly define the spatiotemporal smoothness constraints onthe deformed sequence as the energy functional, which could besolved by gradient descent method. However, due to the high orderderivatives in our energy functional, the computation of partial dif-ferential equation is complex, and the solution space may containmany local optima which can easily trap the algorithm into anundesirable solution.

5. Conclusion

This paper presents a novel nonrigid registration algorithm forcardiac perfusion MR images. Unlike most registration methodsthat estimate the deformation between pairs of images withinthe observed perfusion sequence, we introduce a pseudo groundtruth to facilitate image registration. The pseudo ground truth isa motion-free sequence estimated from the observed perfusionMRI. Since the intensity distributions of the corresponding imagesof the pseudo ground truth sequence and the observed sequenceare almost identical, this method successfully sidesteps the chal-lenges arising from intensity variations during perfusion, andtherefore it need not rely on any multi-modality similarity mea-sure and can improve the performance of many intensity basedregistration methods.

We enhance our previous work in Li and Sun (2009) by usingthe myocardial segmentation information in defining the energyfunctional, a conjugate gradient method in pseudo ground truthfitting, and an edge-emphasized demons method in correspondingframe registration. Therefore, we improve the efficiency and therobustness, which are important in application, of this pseudoground truth based nonrigid registration method.

Compared with other synthetic sequence based approaches(e.g., Buonaccorsi et al., 2005; Adluru et al., 2006; Melbourne


et al., 2007; Milles et al., 2008), our work has the advantages that:(a) it has better generalizability than pharmacokinetic model-based methods; (b) it offers more accurate estimation of perfusionsignals around boundaries thanks to the spatiotemporal smooth-ness constraints; and (c) it has proven to be capable of handlingnonrigid heart motion.

Our experimental results on real patient data have both quanti-tatively and qualitatively shown that our method is able to effec-tively compensate for the elastic deformation of heart, and that itsignificantly outperforms the serial demons registration method(Wang et al., 2005) and an NMI-based FFD approach (Rueckertet al., 1999) for nonrigid registration of myocardial perfusion MRI.

Acknowledgements

The authors thank Siemens Corporate Research, NJ, USA, forproviding the datasets, and acknowledge the support by NUS GrantR-263-000-470-112.

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Pseudo ground truth based nonrigid registration of myocardial perfusion MRI

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