Research Article Nonrigid Medical Image Registration Based...

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2013 Article ID 373082 8 pageshttpdxdoiorg1011552013373082

Research ArticleNonrigid Medical Image Registration Based on MeshDeformation Constraints

XiangBo Lin1 Su Ruan2 TianShuang Qiu1 and DongMei Guo3

1 Faculty of Electronic Information and Electrical Engineering Dalian University of Technology Dalian 116024 China2 Laboratory of Computer Science Information Processing and Systems University of Rouen 76183 Rouen France3 Department of Radiology Second Affiliated Hospital of Dalian Medical University Dalian 116027 China

Correspondence should be addressed to XiangBo Lin linxbodluteducn

Received 1 December 2012 Accepted 4 January 2013

Academic Editor Peng Feng

Copyright copy 2013 XiangBo Lin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Regularizing the deformation field is an important aspect in nonrigid medical image registration By covering the template imagewith a triangular mesh this paper proposes a new regularization constraint in terms of connections between mesh vertices Theconnection relationship is preserved by the spring analogy method The method is evaluated by registering cerebral magneticresonance imaging (MRI) image data obtained from different individuals Experimental results show that the proposed methodhas good deformation ability and topology-preserving ability providing a new way to the nonrigid medical image registration

1 Introduction

Deformable registration is an important tool in medicalimaging with many applications such as atlas-based imagesegmentation and labeling statistical analysis of normal andpathological variations in anatomy and the study of thegrowth anddevelopment of normal and abnormal anatomicalstructures It has been an active research topic for manyyears and plenty of achievements have been published [12] The basic task of image registration is to find a spatialtransformation which maps each point of one image ontoits corresponding point of another image The problem ofimage registration is often considered as a minimizationproblem because it looks for increasing some similaritymetric between the two images to be registered by movingpoints with a reasonable deformation field Common choicesof image similarity metric include sum of squared differ-ences (SSD) normalizedcross-correlation (NCCCC) nor-malizedmutual information (NMIMI) or other divergence-based or information-theoretic measures [3] However itis not sufficient to rely only on similarity metric becausethe solution does not ensure any spatial correlation betweenthe adjacent points Such high dimensional transformationsinvolved in nonrigid registrationmake the problem ill-posedTherefore additional regularizing constraints are required to

enable a reasonable estimation of the displacement field Ingeneral given the target image B and the template image Athe common form of deformable registration problem is

Tlowast = arg minTisinΓ119864 (T) = arg min

TisinΓ119864sim (119861 119860 ∘ T) + 119864reg (T)

(1)

The first term 119864sim(119861 119860 ∘ T) in (1) measures the similaritybetween the deformed template image and the target imageThe second term 119864reg(T) the regularization constraintensures the minimization problem is to be well-posed Theset Γ is the space of admissible transformations The optimaltransformation Tlowast isin Γ is obtained by minimizing theoverall cost function where Γ is the space of admissibletransformations

The regularization constraint plays crucial role in non-rigid registration problem Different constraints have beenproposed in the literature References [4 5] introducedinverse consistency as regularization constraint which canbe explained that the composition of the optimal forwardtransformation and the backward transformation betweenthe template image and the target image is the identitySmooth deformation field is also a commonly used constraint[6 7] against noise Reference [8] used an incompressibilityconstraint ensured by limiting the Jacobian determinant of a

2 Computational and Mathematical Methods in Medicine

transformation should be unity Diffeomorphic transforma-tion is studied much in recent years [9ndash11] that means con-tinuous differentiable and reversible transformation If thetransformation is diffeomorphic the deformed image shouldbe topologically preserved Miller et al [12] proposed thegroup of diffeomorphic mappings for fluid flow registrationThe optimal solution is obtained by regularizing the velocityfield to avoid singular solution in numerical implementationLDDMM[13] aims at finding smooth diffeomorphicmappingfor large deformation by searching the smallest geodesicdistance

Direct topology-preserving constraint is another impor-tant regularization for nonlinear image registrationThemainintuition behind topology preservation in a deformation fieldis the desire to maintain connectivity between neighboringmorphological structures One way to ensure the topologyunchanged during deformation is to keep the Jacobiandeterminant of the transformation always positive The otherway is to identify a credible deformation space accordingto the registration model and then to find the optimaltransformation in the credible deformation space Reference[14] derived elegant linear constraints that provide necessaryand sufficient conditions to ensure that the Jacobian determi-nant values of such transformations are positive everywhereReference [15] extended the work of [14] to 3D B-splinesdeformations They adopted an interval analysis to find themaximum reasonable step along optimal searching path toensure positive Jacobian determinant Reference [16] consid-ered enforcing topology preservation as a hard constraintat several intermediate steps of a deformable registrationprocedure or after the registration was done Reference [17]used a large-scale constrained optimization method to solvethe registration problem By adding conditions involving thegradient of the Jacobian determinant this method encour-aged the topology preserving to be achieved everywhere

Over the past years more and more studies have investi-gated the nonrigid image registration problem with properregularization energies Nevertheless the existing researchis far from being mature There are still different draw-backs Some methods need to track the discrete Jacobiandeterminant or its gradient which significantly increasescomputational cost Some deformable registrationmodels arebuilt in continuous domain Although topology preservationholds for the continuous transformation it is no longerguaranteed when the practical solution is obtained on thediscrete image grid Here we propose a new regularizationscheme for nonrigid image registration to hold topologypreservation The details are organized as follows Section 2illustrates the entire scheme of the proposed nonrigid regis-tration algorithm Section 3 provides the experimental resultsand evaluations of the registration algorithmThe conclusionis drawn in Section 4

2 Methods

Thenonrigid image registrationmodel proposed here focuseson maintaining the unchanged topology of the deformedimage A desirable property of intersubject medical image

warping is the preservation of the topology of anatomicalstructures From medical perspective some normal homol-ogy tissue or structure for any individual such as internalbrain structures should have the same topology A topology-preserving transformation guarantees the unchanged con-nectivity inside a structure and the relationships betweenthe neighboring structures in the deformed image There isno tearing no folding and no appearance or disappearanceof structures To achieve such performance we proposeto cover the deformable template image with a triangularmesh The triangular mesh is generated according to theposition rather than the value of control points Then thetopological relations are controlled by limiting the meshdeformation The advantage is that the algorithm will geta better deformation quality even if no explicit Jacobiandeterminant constraint is used The entire procedure ofNR-MDC is showed in Figure 1 The registration procedureemploys iterative style extraction roughdisplacement field bymeasurements of similarity without considering spatial rela-tions between points and optimization the displacement fieldby measurements of regularization energy A multiresolutionstrategy propagating solutions from coarser to finer scales isused here to speed up the convergence of the algorithm andto avoid local optima

21 Similarity Measures Different features can be used toestablish a similarity metric between the deformed templateimage and the target image to guide the deformed imagetowards the target image [3]Hereweuse SSDas the similaritymetric because it is simple and easy to deal with The SSDforms the basis of the intensity-based image registrationalgorithms and the optimal solution can be obtained byclassical optimization algorithms [18] It can be written as

119864sim (119861 119860 ∘ T) = sumpisinΩ(119861 (p) minus 119860 ∘ T (p))2 (2)

where p is pixel position p = (119909 119910) and Ω is the imagedomain Optical flow field theory [19] is usually adoptedto find the displacement field To better understand themethod we will describe its basic principle roughly Supposean object evolving over time and 119868(p(119905) 119905) is the images ofthis object Based on intensity conservation assumption theimage function satisfies

119868 (p (119905) 119905) = const (3)

By differentiating (3) with respect to t

120597119868

120597119909

120597119909

120597119905

+

120597119868

120597119910

120597119910

120597119905

= minus

120597119868

120597119905

(4)

Consider image 119860 ∘ T and B being two samples of 119868(p(119905) 119905)and let the sampling time to be unit time then

v (p) sdot nabla119861 (p) = 119860 ∘ T (p) minus 119861 (p) (5)

where v(p) = (120597119909120597119905 120597119910120597119905) is the object moving velocity Byapproximation the velocity can be expressed as

v (p) = (119860 ∘ T (p) minus 119861 (p)) nabla119861 (p)nabla119861 (p)2

(6)

Computational and Mathematical Methods in Medicine 3

Template image 119860 Target image 119861

Skull stripping linear preregistration andintensity match

Multiresolution strategy

Displacement field regularization

Figure 1 The framework of the NR-MDC algorithm

Generally the displacement of the point p is u(p) = minusv(p)To avoid unstable solution for small values of nabla119861(p) (6) canbe renormalized to

v (p) = (119860 ∘ T (p) minus 119861 (p)) nabla119861 (p)nabla119861 (p)2 + (119860 ∘ T (p) minus 119861 (p))2

(7)

SSD metric is based on the implicit assumption that theintensities of two corresponding points in images A and Bare equal However this condition is seldom fulfilled in real-world medical image registration because there are many

factors that may affect observed intensity of a tissue overthe imaged field such as the different scanner or scanningparameters normal aging different subjects and so on Inview of such situation intensity normalization is necessaryIn addition small motion assumption should be satisfied inoptical flow field theory To reduce the difference betweenthe template image and the target image as much as possibleit is better to do spatial normalization using rigid or affinetransformation before using (7) to complete the registration

22 Regularization The deformation field obtained fromsimilarity measures does not consider any spatial relationsamong neighboring points The pointsrsquo interdependenciesshould be guaranteed by certain regularization constraintsHere we focus on topology preservation constraints It isone basic topology property that the connection relation-ship between points and edges is unchanged in topologytransformation As is analyzed in [20] three main behaviorsfold cross and tear will cause topological changes duringdeformation To eliminate the unfavorable defects an extratopology correction procedure was done in [20] by trackingthe Jacobian determinant of the deformation field Thuscomputational cost will increase inevitably

The basicmotivation of ourwork is to establish a topologypreservation transformation in discrete domain directly andto avoid calculating the Jacobian determinant from time totime To do that we assume to cover the deformable imagewith a suitablemeshThe central idea of the proposedmethodis to deform the template image in terms of the similaritymetric while controlling the deformation range according tothe intrinsic topologies of the triangular mesh That meansfor each pixel in image level there is a corresponding meshnode to control its motion The crucial problem is to controlmesh deformation perfectly

Research about mesh deformation was done in compu-tational fluid dynamics [21 22] The common applicationsin engineering include deformable aircraft airfoil pneu-matic elastic vibration bionic flow and so on To satisfythe topology preservation transformation requirements weadopt segment spring analogy dynamic mesh technology[23] which is initially used to deform a mesh around apitching airfoil Its basic idea is to replace each mesh edge bya spring Thus the vertex motion will be controlled by all thesprings connected to this vertex

Suppose the mesh edge between two adjacent verticesi and j to be a line spring Initially the mesh is in staticequilibrium state with the spring equilibrium length equalsto the edge length Given an external force along the springthe spring length will change According to Hookrsquos Law theforce at vertex i will be

F119894119895= 119896119894119895(u119895minus u119894) (8)

where 119896119894119895is the spring stiffness between i and j and u

119895is the

displacement of vertex j Influenced by all the neighboringvertices the composition of forces at vertex i is

F119894= sum

119895isin119899119907

F119894119895= sum

119895isin119899119907

119896119894119895(u119895minus u119894) (9)


(a) Before deformation (b) After deformation

Figure 2 The interpretation of the spring analogy

(a) (b) (c)

(d) (e)

Figure 3 BrainMRI images registration results using NR-MDC algorithm (a) Template image (b) target image (c) result without additionalregularization (d) result without updating the template image and (e) result with 3 times updating the template image

where 119899119907is the vertex set whose element connects to vertex

i directly For the mesh to be in equilibrium state again theforce at each vertex should be zero That is

sum

119895isin119899119907

119896119894119895(u119895minus u119894) = 0 (10)

Regrouping (10) yields

u119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119895 (11)

In an iterative style (11) can be rewritten as

u119899+1119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119899119895 (12)

where n denotes the iterative number Then the new positionof vertex i is

x119899+1119894= x119899119894+ u119899+1119894 (13)


60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(a)

60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(b)

Figure 4 The deformation field using NR-MDC algorithm (a) without additional regularization and (b) with additional regularization

The interpretation of the segment spring analogy defor-mation behavior is showed in Figure 2 The green arrowrepresents the vertex displacement vector from previousposition (red dot) to next position (blue dot)

As can be seen from (12) the displacement vector ofvertex i is calculated as a weighted result of the neighboringverticesrsquo displacement The weighting value is determined bythe spring stiffness There are different ways to choose thespring stiffness For simplicity the spring stiffness proposedby Batina [23] is used here which is inversely proportional tothe edge length Consider

119896119894119895=

1

10038161003816100381610038161003816

x119895minus x119894

10038161003816100381610038161003816

(14)

As is analyzed in [24] this spring stiffness diminishes theprobability of vertex collision during the mesh deformationThis means that the factors causing topological changes dur-ing deformation (fold cross and tear) will also be reduced

Introducing segment spring analogy into nonrigid med-ical image registration as a regularization constraint theexpression can be written as

119864reg = sumpisinΩ

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)

where index i represents vertex i whose position is pTherefore the problem is described as

Tlowast = arg minTisinΓsum

pisinΩ(119861 (p) minus 119860 ∘ T (p))2

+ sum

pisinΩ

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(16)

T = x + u (17)

23 Implementation The registration energy function (16) isthe sum of two measures The first term is the sum of scalarwhile the second term is the sum of vector One canminimizethis function with respect to u simultaneously However itis not a trivial work Fortunately alternating minimizationcan be used to approximate the optimal solutions [7 25]First the rough displacement field u is found by minimizingthe similarity metric term Then the optimized displacementfield

simu is found by minimizing the regularization termThis strategy enables the partial minimizations quite fastIn addition to avoid falling into local minima and toreduce the computational cost multiresolution frameworkis adopted The image is downsampled into several differentscales The registration starts from the coarsest scale to thefinest scale Accordingly the resulting deformation transfersfrom the coarsest scale to the finest scale by upsamplingHere the resampling factor is set to be 2 and the imagedimensions of the coarsest scale should not be too small Theprimary algorithm implementation procedure is summarizedas followsAlgorithm Implementation

(1) Given the template image A and the target image Bperform preprocess procedure which includes

(i) skull stripping using the BrainSuite software[26]

(ii) linear spatial normalization using FSL software[27]

(iii) intensity normalization using histogrammatch

(2) Decompose the images into different scales For eachposition at the coarsest scale set the initial displace-ment field u119899 = 0 119899 = 0

(a) compute the transient displacement field uusing (7)


(a) (b) (c)

Figure 5 Typical 3D views for the segmented structures (a) ldquoground truthrdquo segmentation (b) NR-MDC segmentation and (c) DDsegmentation

Table 1 Comparison of the topology preservation ability betweenNR-MDC algorithm and DD algorithm

Algorithm Strategy CC 119873119869

Time (minutes)NR-MDC S1 0971 05 148

S2 0966 0 153S3 0984 0 417

DD 0972 003 541[20] 0979 0 846

(b) get the rough displacement field u119899+1 = u119899 + u

(c) get the regularized displacement fieldsimu119899+1

using(12)

(d) repeat step (a)ndash(c) until convergence(e) transfer to the next finer scale and repeat step

(a)ndash(d) until the finest scale is processed

(3) If there exists a negative Jacobian determinant valueof the final deformation field do 3ndash5 iterations using(12) to optimize the displacement field

(4) If the similarity between the deformed template imageand the target image does not reach the establishedcriteria update the template image using the obtainedtransformation and repeat steps (2)ndash(3)

3 Results

The proposed nonrigid registration approach is evaluated onreal individualrsquos MRI images and compared with Diffeomor-phic Demons algorithm (DD) [28]The reference image usedin the experiment is the one offered by the Surgical PlanningLaboratory of HarvardMedical School [29] It consists of 256times 256 times 160 voxels with a spatial resolution of 09375mm times09375mm times 15mm The test images are real brain MRIimages of fifteen normal subjects provided by the Center forMorphometric Analysis at Massachusetts General Hospitaland available from Internet Brain Segmentation Repository(IBSR) [30]

31 Evaluation of the Topology Preservation Ability Dur-ing the registration both a criterion based on the cross-correlation (noted as CC

119905) and the iteration number con-

stitute the iteration stopping criteria If the cross-correlationbetween the deformed reference image and the target imageequals or exceeds CC

119905or the iteration number of reregistra-

tion using the deformed image as the new reference imagereaches the upper limit (here it is three while within a scalethe iteration number is set to be ten) the algorithm will stopCC119905is defined as

CC119905=

(1 minus CC0)

120572

+ CC0

(18)

where CC0is the initial cross-correlation between the refer-

ence image and the target image The value of 12 is suitablefor parameter 120572 in the experiment

Figures 3 and 4 show some typical registration resultsFigure 3 gives a visual inspection of the registration resultusingNR-MDCalgorithmObviously the deformed templateimage with updating scheme (Figure 3(e)) is more similarto the target image (Figure 3(b)) than the other two results(Figures 3(c) and 3(d)) Figure 4 compares the local deforma-tion field before and after regularization where the red circles(Figure 4(a)) mark out the locations with negative Jacobiandeterminant It can be seen that after several iterations theregularized deformation field becomes realistic

Table 1 compares the NR-MDC algorithm Diffeomor-phic Demons algorithm and method in [20] In Table 1 thecriterion CC (cross-correlation factor) depicts the similaritydegree between the deformed template image and the targetimage CC = 1means the maximum similarity The criterion119873119869depicts the number of points with a negative Jacobian

determinant of the deformation field when compared tothe total point S1 represents the registration result withoutadditional regularization S2 represents the registration resultwithout updating the template image S3 represents theregistration result with regularization and updating stepsSince the vertex motion has high-freedom degree and eachvertex is closely related to its surrounding vertices imme-diate deformation regularization is inadequate There stillexists a small amount of points with changed topologies


Table 2 Comparison of averaged KI value between our NR-MDC algorithm and DD algorithm

L-Caudate R-Caudate L-Putamen R-Putamen L-Thalamus R-Thalamus L-Hippocampus R-HippocampusNR-MDC 0728 0778 0749 0755 0746 0779 0729 0691DD 0710 0762 0732 0741 0722 0754 0711 0682

Thus in most cases additional regularization is necessaryFortunately less iteration is required to achieve acceptabledeformations In addition the need for template imageupdating and the updating number depend on both the initialdifference between the template image and the target imageafter global linear registration and the expected similarityProper updating will improve the similarity The DD algo-rithm is carried out in a multiresolution manner The mainparameters are three scales regularization with a Gaussianconvolution kernel whose standard deviation is set to be onethe maximum template image updating number is three Itcan be seen fromTable 1 that the average CC value is 0984 forNR-MDC algorithm using S3 strategy and the deformationfield is topologically preserved While for the DD algorithmthe average CC value is about 097 and there are still a smallamount of points with negative Jacobian determinants Sincean additional Jacobian determinant tracking procedure wascarried out in themethod of [20] the algorithm running timeis relatively long

32 Evaluation of the Brain Internal Structures SegmentationAbility To further evaluate the reasonability of the obtaineddeformation field the brain internal structures segmentationexperiment is carried out The segmented structures areleft and right caudate (L-Caudate R-Caudate) putamen (L-Putamen R-Putamen) thalamus (L-Thalamus R-Thalamus)and hippocampus (L-Hippocampus R-Hippocampus) Asis known these brain subcortical structures have relativelysmall sizes complex shapes Moreover there is only smallspacing between different structures while their intensities inMRI images are very similar All these negative factors makethe fully automatically accurate segmentation a challengingtask

To validate the results quantitatively a kappa statistic-based similarity index Dice coefficient is adopted in thispaper The similarity index measures the overlap ratiobetween the segmented structure and the ground truthwhich is defined as

KI = 2 times TP2 times TP + FN + FP

(19)

The definitions of the parameters are as follows

TP = 119866 cap 119864 the number of true positiveFP = 119866 cap 119864 the number of false positiveFN = 119866 cap 119864 the number of false negative

where 119866 is the ground truth segmentation of a given struc-ture 119864 is the estimated segmentation of the same structureand 119874 denotes the complement of a set 119874 Perfect spatialcorrespondence between the two segmentations will result inKI = 1 whereas no correspondence will result in KI = 0

The results are presented in Table 2 where the KI valuesare the mean values of all the volumes The results indicatethat the proposed NR-MDC algorithm gives better segmen-tations than the DD algorithm

Typical 3D views for the segmented structures are pre-sented in Figure 5

4 Conclusions

In this paper a new nonrigid medical image registrationmethod named NR-MDC is proposed A new deformablesource a triangular mesh is added to cover the templateimage This mesh is independent of the image intensitiesIt reflects the pointsrsquo intrinsic spatial relations and is usedto regularize the basic deformation field computed from theimage intensity information The proposed cost function isoptimized in an alternative minimization way The deforma-tion field is first computed by optimizing a SSD metric andthen is regularized by using spring analogy method Thisapproach enforces a valid topology of the deformed meshwhich means a valid image deformation To evaluate theperformance of NR-MDC algorithm intersubject brain MRIimage registration experiments are done And a comparativeexperiment is also done between the proposed method andthe state-of-the-art DD algorithm The results verify NR-MDC methodrsquos excellent deformation ability as well as itsgood topology-preserving ability

Acknowledgment

This work was supported by National Science Foundation ofChina (no 61101230)

References

[1] A Gholipour N Kehtarnavaz R Briggs M Devous and KGopinath ldquoBrain functional localization a survey of imageregistration techniquesrdquo IEEE Transactions onMedical Imagingvol 26 no 4 pp 427ndash451 2007

[2] MHolden ldquoA reviewof geometric transformations for nonrigidbody registrationrdquo IEEE Transactions on Medical Imaging vol27 no 1 pp 111ndash128 2008

[3] A Klein J Andersson B A Ardekani et al ldquoEvaluation of 14nonlinear deformation algorithms applied to human brainMRIregistrationrdquo NeuroImage vol 46 no 3 pp 786ndash802 2009

[4] G E Christensen andH J Johnson ldquoConsistent image registra-tionrdquo IEEE Transactions on Medical Imaging vol 20 no 7 pp568ndash582 2001

[5] H J Johnson and G E Christensen ldquoConsistent landmarkand intensity-based image registrationrdquo IEEE Transactions onMedical Imaging vol 21 no 5 pp 450ndash461 2002


[6] D Rueckert L I Sonoda C Hayes D L G Hill M OLeach and D J Hawkes ldquoNonrigid registration using free-form deformations application to breast MR imagesrdquo IEEETransactions onMedical Imaging vol 18 no 8 pp 712ndash721 1999

[7] J PThirion ldquoImagematching as a diffusion process an analogywith Maxwellrsquos demonsrdquo Medical Image Analysis vol 2 no 3pp 243ndash260 1998

[8] T Rohlfing Jr C R Maurer D A Bluemke and M AJacobs ldquoVolume-preserving nonrigid registration of MR breastimages using free-form deformation with an incompressibilityconstraintrdquo IEEE Transactions on Medical Imaging vol 22 no6 pp 730ndash741 2003

[9] PDupuis UGrenander andM IMiller ldquoVariational problemson flows of diffeomorphisms for image matchingrdquo Quarterly ofApplied Mathematics vol 56 no 3 pp 587ndash600 1998

[10] J Ashburner ldquoA fast diffeomorphic image registration algo-rithmrdquo NeuroImage vol 38 no 1 pp 95ndash113 2007

[11] T Vercauteren X Pennec A Perchant and N Ayache ldquoDiffeo-morphic demons efficient non-parametric image registrationrdquoNeuroImage vol 45 no 1 pp S61ndash72 2009

[12] M I Miller A Trouve and L Younes ldquoOn the metrics andEuler-Lagrange equations of computational anatomyrdquo AnnualReview of Biomedical Engineering vol 4 pp 375ndash405 2002

[13] M F Beg M I Miller A Trouve and L Younes ldquoComputinglarge deformation metric mappings via geodesic flows of dif-feomorphismsrdquo International Journal of Computer Vision vol61 no 2 pp 139ndash157 2005

[14] O Musse F Heitz and J P Armspach ldquoTopology preservingdeformable image matching using constrained hierarchicalparametricmodelsrdquo IEEE Transactions on Image Processing vol10 no 7 pp 1081ndash1093 2001

[15] V Noblet C Heinrich F Heitz and J P Armspach ldquo3-Ddeformable image registration a topology preservation schemebased on hierarchical deformationmodels and interval analysisoptimizationrdquo IEEE Transactions on Image Processing vol 14no 5 pp 553ndash566 2005

[16] B Karacalı and C Davatzikos ldquoEstimating topology preservingand smooth displacement fieldsrdquo IEEE Transactions on MedicalImaging vol 23 pp 868ndash880 2004

[17] M Sdika ldquoA fast nonrigid image registration with constraintson the Jacobian using large scale constrained optimizationrdquoIEEE transactions on medical imaging vol 27 no 2 pp 271ndash2812008

[18] X Pennec P Cachier and N Ayache ldquoUnderstanding theldquoDemonrsquos algorithmrdquo 3D non-rigid registration by gradientdescentrdquo in 2nd International Conference on Medical ImageComputing and Computer-Assisted Intervention (MICCAI rsquo99)pp 19597ndash22605 Cambridge UK September 1999

[19] J K Aggarwal and N Nandhakumar ldquoOn the computation ofmotion from sequences of imagesmdasha reviewrdquo Proceedings of theIEEE vol 76 no 8 pp 917ndash935 1988

[20] X Lin T Qiu F Morain-Nicolier and S Ruan ldquoA topologypreserving non-rigid registration algorithm with integrationshape knowledge to segment brain subcortical structures fromMRI imagesrdquo Pattern Recognition vol 43 no 7 pp 2418ndash24272010

[21] D R Mcdaniel and S A Morton ldquoEfficient mesh defor-mation for computational stability and control analyses onunstructured viscous meshesrdquo in Proceedings of the 47th AIAAAerospace Sciences Meeting including the New Horizons ForumandAerospace Exposition AIAAPaper 2009-1363 Orlando FlaUSA January 2009

[22] X Zhou S X Li S L Sun J F Liu and B Chen ldquoAdvances inthe research on unstructured mesh deformationrdquo Advances inMechanics vol 41 pp 547ndash561 2011

[23] J T Batina ldquoUnsteady Euler airfoil solutions using unstructureddynamic meshesrdquo AIAA journal vol 28 no 8 pp 1381ndash13881990

[24] F J Blom ldquoConsiderations on the spring analogyrdquo InternationalJournal for Numerical Methods in Fluids vol 32 pp 647ndash6682000

[25] P Cachier E Bardinet D Dormont X Pennec and NAyache ldquoIconic feature based nonrigid registration the PASHAalgorithmrdquo Computer Vision and Image Understanding vol 89no 2-3 pp 272ndash298 2003

[26] D W Shattuck and R M Leahy ldquoBrainsuite an automatedcortical surface identification toolrdquoMedical Image Analysis vol6 no 2 pp 129ndash142 2002

[27] S M Smith M Jenkinson M W Woolrich et al ldquoAdvances infunctional and structural MR image analysis and implementa-tion as FSLrdquo NeuroImage vol 23 no 1 pp S208ndashS219 2004

[28] T Vercauteren X Pennec A Perchant and N Ayache ldquoDiffeo-morphic demons efficient non-parametric image registrationrdquoNeuroImage vol 45 supplement 1 pp S61ndashS72 2009

[29] R Kikinis M E Shenton D V Losifescu et al ldquoA digital brainatlas for surgical planning model-driven segmentation andteachingrdquo IEEE Transactions on Visualization and ComputerGraphics vol 2 no 3 pp 232ndash241 1996

[30] ldquoInternet brain segmentation repositoryrdquo M G H Techni-cal Report Center for Morphometric Analysis IBSR 2000httpwwwcmamghharvardeduibsr

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transformation should be unity Diffeomorphic transforma-tion is studied much in recent years [9ndash11] that means con-tinuous differentiable and reversible transformation If thetransformation is diffeomorphic the deformed image shouldbe topologically preserved Miller et al [12] proposed thegroup of diffeomorphic mappings for fluid flow registrationThe optimal solution is obtained by regularizing the velocityfield to avoid singular solution in numerical implementationLDDMM[13] aims at finding smooth diffeomorphicmappingfor large deformation by searching the smallest geodesicdistance

Direct topology-preserving constraint is another impor-tant regularization for nonlinear image registrationThemainintuition behind topology preservation in a deformation fieldis the desire to maintain connectivity between neighboringmorphological structures One way to ensure the topologyunchanged during deformation is to keep the Jacobiandeterminant of the transformation always positive The otherway is to identify a credible deformation space accordingto the registration model and then to find the optimaltransformation in the credible deformation space Reference[14] derived elegant linear constraints that provide necessaryand sufficient conditions to ensure that the Jacobian determi-nant values of such transformations are positive everywhereReference [15] extended the work of [14] to 3D B-splinesdeformations They adopted an interval analysis to find themaximum reasonable step along optimal searching path toensure positive Jacobian determinant Reference [16] consid-ered enforcing topology preservation as a hard constraintat several intermediate steps of a deformable registrationprocedure or after the registration was done Reference [17]used a large-scale constrained optimization method to solvethe registration problem By adding conditions involving thegradient of the Jacobian determinant this method encour-aged the topology preserving to be achieved everywhere

Over the past years more and more studies have investi-gated the nonrigid image registration problem with properregularization energies Nevertheless the existing researchis far from being mature There are still different draw-backs Some methods need to track the discrete Jacobiandeterminant or its gradient which significantly increasescomputational cost Some deformable registrationmodels arebuilt in continuous domain Although topology preservationholds for the continuous transformation it is no longerguaranteed when the practical solution is obtained on thediscrete image grid Here we propose a new regularizationscheme for nonrigid image registration to hold topologypreservation The details are organized as follows Section 2illustrates the entire scheme of the proposed nonrigid regis-tration algorithm Section 3 provides the experimental resultsand evaluations of the registration algorithmThe conclusionis drawn in Section 4

2 Methods

Thenonrigid image registrationmodel proposed here focuseson maintaining the unchanged topology of the deformedimage A desirable property of intersubject medical image

warping is the preservation of the topology of anatomicalstructures From medical perspective some normal homol-ogy tissue or structure for any individual such as internalbrain structures should have the same topology A topology-preserving transformation guarantees the unchanged con-nectivity inside a structure and the relationships betweenthe neighboring structures in the deformed image There isno tearing no folding and no appearance or disappearanceof structures To achieve such performance we proposeto cover the deformable template image with a triangularmesh The triangular mesh is generated according to theposition rather than the value of control points Then thetopological relations are controlled by limiting the meshdeformation The advantage is that the algorithm will geta better deformation quality even if no explicit Jacobiandeterminant constraint is used The entire procedure ofNR-MDC is showed in Figure 1 The registration procedureemploys iterative style extraction roughdisplacement field bymeasurements of similarity without considering spatial rela-tions between points and optimization the displacement fieldby measurements of regularization energy A multiresolutionstrategy propagating solutions from coarser to finer scales isused here to speed up the convergence of the algorithm andto avoid local optima

21 Similarity Measures Different features can be used toestablish a similarity metric between the deformed templateimage and the target image to guide the deformed imagetowards the target image [3]Hereweuse SSDas the similaritymetric because it is simple and easy to deal with The SSDforms the basis of the intensity-based image registrationalgorithms and the optimal solution can be obtained byclassical optimization algorithms [18] It can be written as

119864sim (119861 119860 ∘ T) = sumpisinΩ(119861 (p) minus 119860 ∘ T (p))2 (2)

where p is pixel position p = (119909 119910) and Ω is the imagedomain Optical flow field theory [19] is usually adoptedto find the displacement field To better understand themethod we will describe its basic principle roughly Supposean object evolving over time and 119868(p(119905) 119905) is the images ofthis object Based on intensity conservation assumption theimage function satisfies

119868 (p (119905) 119905) = const (3)

By differentiating (3) with respect to t

120597119868

120597119909

120597119909

120597119905

+

120597119868

120597119910

120597119910

120597119905

= minus

120597119868

120597119905

(4)

Consider image 119860 ∘ T and B being two samples of 119868(p(119905) 119905)and let the sampling time to be unit time then

v (p) sdot nabla119861 (p) = 119860 ∘ T (p) minus 119861 (p) (5)

where v(p) = (120597119909120597119905 120597119910120597119905) is the object moving velocity Byapproximation the velocity can be expressed as

v (p) = (119860 ∘ T (p) minus 119861 (p)) nabla119861 (p)nabla119861 (p)2

(6)









(7)







F119894119895= 119896119894119895(u119895minus u119894) (8)


119895is the


F119894= sum

119895isin119899119907

F119894119895= sum

119895isin119899119907

119896119894119895(u119895minus u119894) (9)




(a) (b) (c)

(d) (e)




sum

119895isin119899119907

119896119894119895(u119895minus u119894) = 0 (10)


u119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119895 (11)


u119899+1119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119899119895 (12)


x119899+1119894= x119899119894+ u119899+1119894 (13)


60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(a)

60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(b)




119896119894119895=

1

10038161003816100381610038161003816

x119895minus x119894

10038161003816100381610038161003816

(14)




10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)



pisinΩ(119861 (p) minus 119860 ∘ T (p))2

+ sum

pisinΩ

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(16)

T = x + u (17)










(a) (b) (c)





S2 0966 0 153S3 0984 0 417

DD 0972 003 541[20] 0979 0 846



using(12)





3 Results







CC119905=

(1 minus CC0)

120572

+ CC0

(18)













(19)






4 Conclusions


Acknowledgment


References





































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
























(7)







F119894119895= 119896119894119895(u119895minus u119894) (8)


119895is the


F119894= sum

119895isin119899119907

F119894119895= sum

119895isin119899119907

119896119894119895(u119895minus u119894) (9)




(a) (b) (c)

(d) (e)




sum

119895isin119899119907

119896119894119895(u119895minus u119894) = 0 (10)


u119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119895 (11)


u119899+1119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119899119895 (12)


x119899+1119894= x119899119894+ u119899+1119894 (13)


60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(a)

60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(b)




119896119894119895=

1

10038161003816100381610038161003816

x119895minus x119894

10038161003816100381610038161003816

(14)




10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)



pisinΩ(119861 (p) minus 119860 ∘ T (p))2

+ sum

pisinΩ

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(16)

T = x + u (17)










(a) (b) (c)





S2 0966 0 153S3 0984 0 417

DD 0972 003 541[20] 0979 0 846



using(12)





3 Results







CC119905=

(1 minus CC0)

120572

+ CC0

(18)













(19)






4 Conclusions


Acknowledgment


References





































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















(a) (b) (c)

(d) (e)




sum

119895isin119899119907

119896119894119895(u119895minus u119894) = 0 (10)


u119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119895 (11)


u119899+1119894=

1

sum119895isin119899119907

119896119894119895

sum

119895isin119899119907

119896119894119895u119899119895 (12)


x119899+1119894= x119899119894+ u119899+1119894 (13)


60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(a)

60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(b)




119896119894119895=

1

10038161003816100381610038161003816

x119895minus x119894

10038161003816100381610038161003816

(14)




10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)



pisinΩ(119861 (p) minus 119860 ∘ T (p))2

+ sum

pisinΩ

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(16)

T = x + u (17)










(a) (b) (c)





S2 0966 0 153S3 0984 0 417

DD 0972 003 541[20] 0979 0 846



using(12)





3 Results







CC119905=

(1 minus CC0)

120572

+ CC0

(18)













(19)






4 Conclusions


Acknowledgment


References





































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(a)

60 65 70 75 80 8580

85

90

95

100

105

110

115

120

(b)




119896119894119895=

1

10038161003816100381610038161003816

x119895minus x119894

10038161003816100381610038161003816

(14)




10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(15)



pisinΩ(119861 (p) minus 119860 ∘ T (p))2

+ sum

pisinΩ

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895isin119899119907

119896119894119895(u119895minus u119894(p))10038161003816100381610038161003816100381610038161003816100381610038161003816

(16)

T = x + u (17)










(a) (b) (c)





S2 0966 0 153S3 0984 0 417

DD 0972 003 541[20] 0979 0 846



using(12)





3 Results







CC119905=

(1 minus CC0)

120572

+ CC0

(18)













(19)






4 Conclusions


Acknowledgment


References





































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















(a) (b) (c)





S2 0966 0 153S3 0984 0 417

DD 0972 003 541[20] 0979 0 846



using(12)





3 Results







CC119905=

(1 minus CC0)

120572

+ CC0

(18)













(19)






4 Conclusions


Acknowledgment


References





































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of























(19)






4 Conclusions


Acknowledgment


References





































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















Research Article Nonrigid Medical Image Registration Based...

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