This paper presents a novel technique for a consistent symmetric deformable image registration based on an accurate method for a direct inversion of a large motion model deformation field. The proposed image registration algorithm maintains one-to-one mapping between registered images by symmetrically warping them to another image. This makes the final estimation of forward and backward deformation fields anatomically plausible and applicable to adaptive prostate radiotherapy. The quantitative validation of the method is performed on magnetic resonance data obtained for pelvis area. The experiments demonstrate the improved robustness in terms of inverse consistency error and estimation accuracy of prostate position in comparison to the previously proposed methods

Matuszewski, Bogdan J

Papiez, Bartek

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PAPIEZ, MATUSZEWSKI: PELVIC-AREA SYMMETRIC IMAGE REGISTRATION 1Direct inverse deformation field approach topelvic-area symmetric image registrationBartlomiej W. Papiezbwpapiez@uclan.ac.ukBogdan J. Matuszewskibmatuszewski1@uclan.ac.ukADSIP Research CentreUniversity of Central LancashirePreston, UKAbstractThis paper presents a novel technique for a consistent symmetric deformable imageregistration based on an accurate method for a direct inversion of a large motion modeldeformation field. The proposed image registration algorithm maintains one-to-one map-ping between registered images by symmetrically warping them to another image. Thismakes the final estimation of forward and backward deformation fields anatomicallyplausible and applicable to adaptive prostate radiotherapy. The quantitative validationof the method is performed on magnetic resonance data obtained for pelvis area. Theexperiments demonstrate the improved robustness in terms of inverse consistency errorand estimation accuracy of prostate position in comparison to the previously proposedmethods.1 IntroductionImage registration is a fundamental task in medical image processing aiming at an optimal,in some sense, estimation of spatial transformation aligning two or more images. As the im-age registration is an ill-posed problem it needs to be regularised by introducing additionala priori information to the estimation process [4]. In the classical formulation of a non-parametric image registration, methods based on elastic, fluid, diffusive deformable models[5] are commonly used to enforce a globally smooth dense deformation field. Althoughthose methods have been shown to be fast and accurate, they have a drawback when usedin clinical applications as they do not explicitly preserve organs’ topology. To maintain theneighbourhood relationship and avoid anatomically incorrect deformations, the inverse con-sistency error (ICE) has been introduced. In the earliest work on minimising the ICE duringimage registration, an algorithm jointly estimating a forward and a backward transformationwas proposed [2]. A similar idea of simultaneously reducing the ambiguous correspondencebetween the forward and the backward transformation but established via a variational frame-work and not limited to mono-modal images was presented in [8]. Recently a diffeomorphicformulation of the image registration was proposed [1, 6] as an efficient way of preventingtransformation folding. All of those methods have been validated on MRI and CT imagesof a brain with relatively small deformations, meanwhile the adaptive radiotherapy (ART) ofprostate cancer requires to cope with significant changes of bladder and rectum shape andsize. To overcome this problem, a symmetric warping between two images was introducedc© 2011. The copyright of this document resides with its authors.It may be distributed unchanged freely in print or electronic forms.2 PAPIEZ, MATUSZEWSKI: PELVIC-AREA SYMMETRIC IMAGE REGISTRATIONFigure 1: Symmetric image registration schemeby registering these images to an intermediate image [3, 7]. These methods require thoughan explicit calculation of the inverse deformation fields.The method proposed in this paper extends the approach presented in [7] by directlyinverting deformation field in each iteration. This allows the alleviation of constraints im-posed on the maximum magnitude of the deformation field update in every iteration of thealgorithm. Finally, the proposed registration scheme is compared against the algorithm pro-posed in [7], demonstrating improvement of the ICE and the accuracy of the prostate positionestimate.2 Symmetric Image RegistrationThe consistent symmetric image registration is defined here for mono-modal images with Arepresenting a fixed (reference) image and B a moving image. Corresponding deformation(displacement) fields at any spatial position are defined as: ~x: TAC =~x+~u(~x) and TBC =~x+~v(~x) warping respectively image A and image B to an intermediate imageC. Mathematicallyit can be stated as an optimisation problem:arg minu,v(Sim(A◦u,B◦ v)+αuReg(u)+αvReg(v)) (1)where: Sim is a chosen similarity measure between images (e.g. the Sum of Squared Differ-ences [5]), Reg is a regularisation term, and αu, αv are regularisation weights. To solve thisproblem, the Demon-like force established in an iterative optimisation framework [6, 7] waschosen:dui+1 =(Ai−Bi)(∇Ai+∇Bi)‖∇Ai+∇Bi‖2 +(Ai−Bi)2 (2)where: Ai is warped image A using estimated deformation field ui; Bi is warped image Busing estimated deformation field vi; ∇Ai is gradient of image Ai; ∇Bi is gradient of image Bi;i is an index of the current iteration. The results of this registration towards the intermediateimage: TAC and TBC need to be inverted and the final transformations TAB and TBA are thecompositions of TAC and TBC and their inverses T−1AC and T−1BC : TAB = TAC ◦T−1BC and TBA =TBC ◦T−1AC . The overall scheme of this registration process is illustrated in Fig. 1.2.1 Small-step multiple pass approachIn a small-step multiple pass approach originally proposed in [7], it is assumed that:ui+1 = Ge ∗(ui ◦(G f ∗ (dui)))vi+1 = Ge ∗(vi ◦(G f ∗ (−dui)))(3)where: Ge∗ and G f ∗ represent Gaussian kernel convolutions which operate on updated dis-placement fields u and v and updated velocity field du respectively. This assumption sim-plifies significantly the estimation of the deformation fields but it holds only for small up-dates. The Demon-like force does not guarantee the small-step update and therefore thePAPIEZ, MATUSZEWSKI: PELVIC-AREA SYMMETRIC IMAGE REGISTRATION 3explicit procedure, limiting the deformation magnitude is applied when the estimated updateis greater than 0.4 voxel size. However, image registration in the ART requires not only tobe accurate but also fast and the update magnitude limiting procedure contradicts these re-quirements. Additionally it needs to be checked how good this approximation of the inverseupdate is in practise.2.2 Direct inverse deformation field approachThe proposed method builds on the previous approach by directly inverting update of thedeformation field in each iteration. The directly inverted deformation field is obtained byfinding zero of a misalignment function ~f (~x) defined for an arbitrary point~y form the rangeof transformation ~Tinput : ~f (~x) =~y−~Tinput(~x). It can be shown using Taylor expansion, that~f (~x+ ~dx) ≈ ~f (~x) + J(~f (~x))~dx. The estimated ~x can be found in an iterative manner; foriteration i, assuming that ~f (~xi+ ~dxi) = 0, ~xi+1 is given by ~xi+1 =~xi+ ~dxi where ~dxi is cal-culated by solving a set of linear equations: J(~f (~x)+ β I)~dx = ~f (~x). Here J(~f (~x)) is theJacobian of ~f (~x). The β I, where I is the identity matrix and β is a non-negative numberis used to regularise the set of equations when the condition number of J(~f (~x)) is above agiven threshold, otherwise β is set to zero. This method of inverting deformation field canbe seen as a modification of the method proposed in [2], enabling to invert more accuratelylarge deformations. Finally, the corresponding update scheme is as follows:ui+1 = Ge ∗(ui ◦(G f ∗ (dui)))vi+1 = Ge ∗(vi ◦(G f ∗(du−1i)))(4)This scheme in contrast to Eq. 3 (compare formulas for updating vi+1) uses the direct in-verse update of the deformation which does not suffer from the limitations of the small-stepmultiple pass approach.3 Experimental resultsTo evaluate the accuracy and robustness of the proposed approach, multiple pelvic MRIscans of the same subject were used. For quantitative evaluation, the previously proposedframework [7] was compared against the proposed method using the inverse consistencyerror (ICE) measure defined as:ICE(TAB,TBA)(~x) =12(‖(~x− (TAB ◦TBA)(~x))‖+‖(~x− (TBA ◦TAB)(~x))‖) (5)and maximal ICE (maxICE):maxICE(TAB,TBA)(~x) = max(‖(~x− (TAB ◦TBA)(~x))‖,‖(~x− (TBA ◦TAB)(~x))‖) (6)The relative overlap (RO) between prostate segmented in the reference image, Pre f , andin the warped moving image after registration, Pwarp, was used to evaluate the registrationperformance in terms of the prostate position. The RO has been defined as:RO(Pre f ,Pwarp) =2∗numberO fVoxels(Pre f ∩Pwarp)numberO fVoxels(Pre f ∪Pwarp) (7)The data set consists of 5 volumes of 320x240x30 voxels with voxel size of 1.0x1.0x3.0mm.In each scan, the data exhibit significant changes of bladder size and shape. A sample of the4 PAPIEZ, MATUSZEWSKI: PELVIC-AREA SYMMETRIC IMAGE REGISTRATIONFigure 2: Example of the data used in the experiments: volume labelled as Image 3, selectedas the reference image (left); volume labelled as Image 5 (right). Segmented bladder, rectumand prostate are shown in red, green and blue respectively.Figure 3: Axial (top), coronal (middle) and sagittal (bottom) views of the difference betweenreference image and moving image before registration (left), after registration using Yang’smethod (middle), and the proposed method (right).data is shown in Fig. 2. Image which is labelled as Image 3 was chosen as a reference imagefor all experiments.The dispersion of 0.5 and 1.0 was used for Ge and G f smoothing Gaussian kernels re-spectively; the maximal number of iteration was set to 50 and the multi-linear interpolationmethod was implemented to estimate non-grid values of images and deformation fields.The results presented in Tab. 1 show that the proposed framework produces smaller ICEand maxICE than previously proposed method, especially when a significant deformationneeds to estimated (e.g.Image 4 and Image 5). The difference between images before andafter registration is shown in Fig. 3. The proposed method is also seen as more robust.This is mainly due to applied method for the direct calculation of the inverse deformationfield. The proposed method performs better when compared to Yang’s method [7] in termsof prostate position accuracy, as in all the cases the proposed method achieve greater valuesof RO as illustrated in Fig. 4.4 SummaryRegistration of pelvic area images is challenging due to possible significant shape and sizechanges of bladder and rectum. To provide an accurate method for estimation of the prostateposition, the consistent symmetric image registration framework based on the direct inver-sion procedure is proposed in this paper. The quantitative validation preformed on real MRIdata shows that proposed modifications of the previously reported algorithms resulted inPAPIEZ, MATUSZEWSKI: PELVIC-AREA SYMMETRIC IMAGE REGISTRATION 5ICE (maxICE)Image 1 Image 2 Image 4 Image 5Yang’s [7] 0.27 (8.08) 0.14 (5.07) 0.14 (4.96) 0.34 (10.8)The proposed 0.04 (1.68) 0.04 (1.30) 0.03 (0.96) 0.05 (3.84)Table 1: ICE and maxICE (in brackets) calculated for deformation field estimated usingYang’s framework [7] and the proposed framework.Figure 4: RO for segmented prostate before registration (blue), after registration using Yang’smethod [7] (red), and after registration using the proposed method (yellow).the somewhat improved prostate RO measure and the significant reduction of the ICE andmaxICE measures.AcknowledgementThe work was supported by the EPSRC (grant number EP/D077540/1).References[1] J. Ashburner. A fast diffeomorphic image registration algorithm. NeuroImage, 38(1):95–113, 2007.[2] G.E. Christensen and H.J. Johnson. Consistent image registration. IEEE Transactionson Medical Imaging, 20(7):568–582, 2001.[3] X. Han, L.S. Hibbard, and V. Willcut. An efficient inverse-consistent diffeomorphicimage registration method for prostate adaptive radiotherapy. In Proc. MICCAI 2010Workshop on Prostate Cancer Imaging, pages 34–41. Springer-Verlag, 2010.[4] B.J. Matuszewski, J.K. Shen, L.K. Shark, and Moore C.J. Estimation of internal bodydeformations using an elastic registration technique. In Proc. MedVis, pages 15–20.IEEE, 2006.[5] J. Modersitzki. FAIR: Flexible Algorithms for Image Registration. SIAM, 2009.[6] T. Vercauteren, X. Pennec, A. Perchant, and N. Ayache. Diffeomorphic demons: Effi-cient non-parametric image registration. NeuroImage, 45(1, Supp.1):61–72, 2009.[7] D. Yang, H. Li, D. A. Low, J. O. Deasy, and I. El Naqa. A fast inverse consistentdeformable image registration method based on symmetric optical flow computation.Physics in Medicine and Biology, 53(21):6143, 2008.[8] Z. Zhang, Y. Jiang, and H. Tsui. Consistent multi-modal non-rigid registration based ona variational approach. Pattern Recogn. Lett., 27:715–725, 2006.

Direct inverse deformation field approach to pelvic-area symmetric image registration

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