Image Registration via Particle Movement

Transcription

1 Image Registration via Particle Movement Zao Yi and Justin Wan Abstract Toug fluid model offers a good approac to nonrigid registration wit large deformations, it suffers from te blurring artifacts introduced by te viscosity term. To overcome tis drawback, we present an inviscid model expressed in a particle framework. Our idea is to simulate te template image as a set of particles moving towards te target positions. Te proposed model can accommodate bot small and large deformations wit sarp edges and reasonable displacement maps acieved. Because of its simplicity, te computational cost is muc smaller tan oter nonrigid registration approaces. 1 INTRODUCTION Image registration requires aligning an image pair wit an optimal transformation. It as many potential applications for diagnosing in te clinic, and is an often encountered problem in various fields especially in medical surgery. It is well known tat biological structures suc as uman brains, altoug may contain te same global structures, differ in sape, orientation, and fine structures across individuals and at different times. To represent suc variabilities, nonrigid transformation as more applications [10]. Numerous algoritms ave been explored in tis area, owever, more accurate and efficient metods are still needed. Broit [3] was te first to study nonrigid registration problems using pysically based models. In is approac te transformation process was modeled by deformations of an elastic solid. Tis approac was extended by Bajcsy et al. [1] and various variational forms [4, 8] were proposed later. Since te deformation energy caused by stress increases proportionally wit te strengt of te deformation, elastic models can only accommodate locally small deformations. To overcome tis drawback, Cristensen et al. [6] proposed anoter approac wic modeled te transformation process by a viscous fluid flow. Fluid models can allow relatively larger local deformations; owever, te inerent viscosity introduces an unnegligible blurring effect to te transformed image. Also, te reported [2, 12] computational time is very large. Te models discussed above are all pysically based models, wic try to simulate te deformation as a pysical process. Since te viscosity term in te fluid equation causes blurring as well as complicating computation, we propose to use an inviscid model. Instead of regarding te template image as a fluid continuum wit viscous interaction, we 1

2 simulate te transformation process as a set of particles moving towards target positions. It is similar to te framework of gas dynamics [9] were distances between gas molecules are big enoug tat internal friction can be neglected and eac molecule can be viewed as a separate particle. Particles are by far te easiest objects to simulate. Despite teir simplicity, particles can be made to exibit a wide range of interesting beaviors. For example, in grapics [7] a variety of nonrigid structures can be built by connecting particles wit simple damped springs. However, te particle approac as never been used in image registration area. Based on te pysical beavior of particles, we present a novel registration tecnique expressed in a particle framework. Te basic idea is to simulate te template image as particles moving towards te target image under external forces. Lagrangian reference frame is often used to observe te movement of particles. However, it needs to track individual particle and is arder to apply. As time varies, initially equally-sized particles may become unevenly distributed wic can cause problems of stability and accuracy. Assuming tere are enoug particles moving around, we can fit Eulerian reference frame to te particle registration framework. Te resulting partial differential equations are nonlinear yperbolic equations of vector form wose solution describes te coordinate transformation between te template and te target images. Tey can be numerically solved using finite difference metod. Te proposed particle registration tecnique is quite simple and efficient. Since it can be viewed as an inviscid gas model wit constant pressure, te blurring artifacts caused by fluid viscosity are overcome and large deformation can be accommodated. Also, because of its simplicity in simulation, total computational cost is decreased. Tus, te particle registration tecnique can acieve a small/large transformed image wit sarp edges and reasonable displacement map in less time. We ave successfully applied te particle model to medical image datasets yielding fast and accurate registrations. A relevant approac was te level-set registration tecnique proposed by Vemuri el al. [11]. As an inviscid model, teir approac was derived from curve evolution teory wic ad different settings and governing equations, and tus could not be regarded as a particle framework. Te rest of te paper is organized as follows. Section 2 contains an overview of te particle model, followed by te proposed formulation of te image registration problem. Section 3 presents some modifications of te model regarding robustness and stability, and gives te wole numerical algoritm for solving te consequent equations. Section 4 sows experimental results of tis approac on 2D sytetic/real images. Finally, conclusions are drawn in Section 5. 2 METHODOLOGY We define te template image as I 1 (x) and te target image as I 2 (x), were 0 I 1 (x), I 2 (x) 1, and x Ω is te image region. Te purpose of image registration is to determine a coordinate transformation r(x) of I 1 (x) onto I 2 (x) 2

3 suc tat te difference between te transformed template image I 1 (x r(x)) and te target image I 2 (x) will be small in some measure M. Terefore te registration can also be stated as a minimization problem min M(I 1(x), I 2 (x), r(x)). (1) r(x) Here we use a Gaussian sensor model [5] to induce te matcing criteria: M(I 1 (x), I 2 (x), r(x, t)) = α I 1 (x r(x, t)) I 2 (x) 2 dx, (2) 2 Ω were α is a parameter. We consider te template image as a set of particles, eac wit mass m, position x, velocity u, displacement r, and responding to force. Te image deformation process is simulated as te particles moving towards te target positions. Te total number of particles is assumed to be big enoug suc tat at anytime tere is a particle passing troug eac observation point. Tus, we can use Eulerian reference frame for simulation. Te image region Ω is discretized into a fixed grid and motion of particles is controlled in eac cell. Te velocity field u(x) and te displacement field r(x) are bot defined based on current positions of particles. A particle at position x at time t originated at position x r(x, t). Te movement of a particle is governed by Newton s Law of Motion. If we assume tat tere is no internal interaction between particles, te conservation of momentum equation can be written as m du dt = f, (3) were f is te external force applied to tat particle and will be defined later by information from te template and te target images, m is te mass of eac particle wic is assumed to be a constant ere, u is te consequent velocity wic describes te speed of image deformation, and t is te time. In an Eulerian framework, we ave te following relationsip between total derivative and partial derivative of velocity wit respect to time du dt = u + (u )u, (4) t were is te gradient operator. Tus, equation (3) can be rewritten as m u t + m(u )u = f. (5) Te left side represents te force of inertia, i.e., te mass times te acceleration of a particle. Equation (5) neglects internal friction and terefore is an inviscid model. It is similar to te Euler equation commonly used in gas dynamics (te study of compressible but inviscid fluids) ρ u t + ρ(u )u + p = f, (6) 3

4 were ρ is te density or unit mass, and p is te pressure. For isotermal gas wit constant density, p is neglected and equation (6) becomes equation (5). Since velocity is te total derivative of displacement wit respect to time, te velocity field and te displacement field are related by u(x, t) = dr(x, t) dt = r(x, t) t + r(x, t)u(x, t). (7) Te force field f(x, r(x)) is used to drive te particles from image I 1 (x) to image I 2 (x). It is defined as te derivative of te matcing criteria M on te image pair. Taking te variation of equation (2) wit respect to displacement, r(x, t), gives te force field f(x, r(x, t)) = α (I 1 (x r(x, t)) I 2 (x)) I 1 (x r(x, t)). (8) Since te movement of particles sould only slow down wen te difference term I 1 (x r(x, t)) I 2 (x) becomes smaller, we need to normalize te gradient term in te force field. Terefore we coose α = α I 1 suc tat equation (8) becomes f(x, r(x, t)) = α(i 1 (x r(x, t)) I 2 (x)) I 1(x r(x, t)) I 1 (x r(x, t)). (9) 3 IMPLEMENTATION Solution of te particle registration problem requires solving te following PDEs u(x, t) f(x, r(x, t)) = (u )u(x, t), t m r(x, t) = u(x, t) r(x, t)u(x, t), t f(x, t) = α(i 1 (x r(x, t)) I 2 (x)) I 1(x r(x, t)) I 1 (x r(x, t)), wic include nonlinearity introduced by te external force and te kinematic derivatives. Two issues arise in calculating te external force field. Te first is te undetermination of external force wen I 1 (x r(x, t)) equals to zero. To make our model more robust, we apply an additive external force definition in tat case: f(x, r(x, t)) = 0. (10) Te second is te computation of gradient I 1 (x r(x, t)). Since te gradient operator is sensitive to te noise in te image, we convolve I 1 (x r(x, t)) wit a Gaussian kernel prior to te gradient computation. Tus (9) and (10) become { 0, if (Gσ I(x, t)) = 0; f(x, r(x, t)) = α(i(x, t) I 2 (x)) (Gσ I(x,t)) (G σ I(x,t)), oterwise; (11) 4

5 were I(x, t) = I 1 (x r(x, t)) denotes te deformed template image at time t, G σ denotes te Gaussian kernel wit standard deviation of σ, and is te convolution operator. To update movement of particles troug time, we discretize time domain [0, + ] into small intervals 0 = t 0 < t 1 <... < t n < t n+1 <... and apply Euler explicit integration over time. Te discretized time formula is given by were u n+1 (x) = u n (x) + t m f n (x), r n+1 (x) = r n (x) + t(i r n (x))u n (x), (12) u n+1 (x) u(x, t n+1 ) is te approximation of velocity field at time t n+1, r n+1 (x) r(x, t n+1 ) is te approximation of displacement field at time t n+1, t = t n+1 t n is te time interval. We require te Jacobian J = I r n [6] greater tan zero in order to obtain a regular transformation. Te complete algoritm for solving te particle registration problem consequently becomes 1. Initialize u 0 (x) = 0 and r 0 (x) = Calculate te external force f n (x) at time t n using equation (11). 3. If f n (x) is below a stopping criteria for all x or te maximum number of iterations is reaced, STOP. 4. Perform Euler explicit integration using equation (12) at time t n. 5. If te Jacobian J = I r n (x) is less tan 0.1, regrid te template. 6. n = n + 1, GOTO step 2. Te remaining question is ow to perform steps 2, 4, and 5 in discretized space domain. Note tat tey all ave gradient computation included. We coose minmod finite difference sceme since it is known to preserve local max/min and yield acceptable accuracy. Te minmod function [11] is defined as { sign(x) min( x, y ), if xy > 0; minmod(x, y) = (13) 0, if xy 0; and consequently te gradient is computed by [ minmod(d F (x, y) = x F, D x + F ) minmod(dy F, D y + F ) ], (14) were F (x, y) is any 2D function, D x +, Dx, and D y +, Dy are standard forward and backward difference operators in te x and y directions, respectively. 5

6 To update movement of particles troug space, we discretize space domain Ω into small cells/pixels. Eac particle is initially located at te center of te cell/pixel wit position x ij = (x i, y j ) and grid spacing. Applying minmod finite difference over space, we obtain te discretized time discretized space formula fij n = α(in ij I 2,ij) minmod( I n ij In i 1,j minmod( In ij In i 1,j gij n = α(in ij I 2,ij) minmod( I ij I i,j 1 u n+1 v n+1 ij = u n ij + tf ij n, ij = vij n + tgn ij, r n+1 ij s n+1 ij were minmod( In ij In i,j 1 = r n ij + t((1 minmod( rn ij rn i 1,j = s n ij + t((1 minmod( sn ij sn i,j 1, In i+1,j In ij ), In i+1,j In ij ), I i,j+1 I ij ), In i,j+1 In ij ),,, rn i+1,j rn ij, sn i,j+1 sn ij ))u n ij minmod( rn ij rn i,j 1, rn i,j+1 rn ij ))v n ij minmod( sn ij sn i 1,j )vij n ),, sn i+1,j sn ij )u n ij ), (fij n, gn ij ) f(x ij, r(x ij, t n )) is te approximation of force at position x ij time t n, (u n+1 ij, v n+1 ij ) u(x ij, t n+1 ) is te approximation of velocity at position x ij time t n+1, (r n+1 ij, s n+1 ij ) r(x ij, t n+1 ) is te approximation of displacement at position x ij time t n+1, t = t n+1 t n is te time interval, Iij n = G σ I 1,xi rij n,y j s n is te smooted deformed template. ij Te discretization is easily extendable to 3D. 4 RESULTS Te proposed approac is implemented in C and executed at a desktop PC of P4 2.8GHZ wit 1GB memory. Te parameters are set to m = 1, α = 100, and te stopping criteria is set to We ave applied te proposed algoritm to tree experiments. Four sets of images are presented for eac registration: 1. template image; 2. target image; 3. transformed image after registration; 4. difference image between transformed image and target image. Te computational cost for eac experiment is summarized in Table 1. Te first experiment is designed to demonstrate tat our model can accommodate large deformation like fluid algoritm as well as acieving a reasonable displacement map. Te test data is a syntetic image wit size pixels wic is similar to te image used by [6]. Te results are sown in Figure 1 wit comparison to fluid model. It is wat we desire tat te transformed image Time(s) experiment 1 experiment 2 experiment 3 Fluid Proposed Table 1: computational cost for eac experiment 6

7 Figure 1: qualitative results of experiment 1. From left to rigt, top to bottom: (a)template image; (b)target image; (c)transformed image obtained by fluid model; (d)difference image between (b) and (c); (e)transformed image obtained by proposed model; (f)difference image between (b) and (f); (g)displacement map obtained by fluid model; ()displacement map obtained by proposed model. obtain by proposed model is almost te same as te target, and te difference image is better tan tat obtained by fluid model. Besides te four sets of images mentioned above, we include displacement map as well. For proposed model, it is largely curved from a small patc of letter C to te wole letter C, and anywere else is zero. Tis contrasts wit te displacement map obtained by fluid model were background field also as nonzero displacements. Te second experiment is designed to demonstrate tat our model can overcome te blurring drawback of fluid approac. Te test data is a syntetic image wit size pixels wic is similar to te image used by [12]. Te results are sown in Figure 2 wit comparison to fluid model. Again we include displacement map in addition to te four sets of images mentioned above. From comparison we can see clearly tat te difference image obtained by fluid model is more blurry tan tat obtained by te proposed model. Tus, we demonstrate tat our model acieves a transformed image wit clearer texture tan te fluid approac. 7

8 Figure 2: qualitative results of experiment 2. From left to rigt, top to bottom: (a)template image; (b)target image; (c)transformed image obtained by fluid model; (d)difference image between (b) and (c); (e)transformed image obtained by proposed model; (f)difference image between (b) and (f); (g)displacement map obtained by fluid model; ()displacement map obtained by proposed model. 8

9 Figure 3: qualitative results of experiment 3. From left to rigt, top to bottom: (a)template image; (b)target image; (c)transformed image obtained by fluid model; (d)difference image between (b) and (c); (e)transformed image obtained by proposed model; (f)difference image between (b) and (f). Te tird experiment is designed to demonstrate tat our model can capture complex deformations in medical images. Te test data is an MRI image of uman brain wit size pixels. Te results are sown in Figure 3. Te registration as successfully transformed te template image towards te target image, and te transformed image persists sarp edges. To furter assess te quality of te registration in te above experiments, mean and variance of te squared sum of intensity difference (SSD), and correlation coefficient (CC) ave been calculated and are listed in tables 2. 5 CONCLUSIONS In tis paper we present a novel registration tecnique expressed in a particle framework. It is an inviscid model designed for nonrigid registration problems. Te key features of our model are (a) it can accommodate bot small and large deformations, (b) it overcomes te blurring drawback of fluid models and acieves transformed images wit clear textures and sarp edges, (c) it is very simple and fast. We ave demonstrated te performance of our approac on a variety of images including syntetic and real data. Te results of experiments are desiring and satisfying. Future efforts will be made to explore more complicated simulation frameworks were internal interaction is added to particle systems. 9

10 References Experiment 1 Mean(SSD) Var(SSD) CC No registration Fluid Proposed Experiment 2 Mean(SSD) Var(SSD) CC No registration Fluid Proposed Experiment 3 Mean(SSD) Var(SSD) CC No registration Fluid Proposed Table 2: qualitative measure for 3 experiments [1] R. Bajcsy and S. Kovacic. Multiresolution elastic matcing. Computer Vision, Grapics, and Image Processing, 46:1 21, [2] M. Bro-Nielsen and C. Gramkow. Fast fluid registration of medical images. In Proc. Visualization in Biomedical Computing, pages , [3] C. Broit. Optimal registration of deformed images. PD tesis, University of Pennsylvania, [4] G.E. Cristensen, S.C. Josi, and M.I. Miller. Volumetric transformation of brain anatomy. IEEE Trans. Medical Imaging, 16: , [5] G.E. Cristensen, R.D. Rabbitt, and M.I. Miller. A deformable neuroanatomy textbook based on viscous fluid mecanics. In Proc. Information Science and Systems, pages , [6] G.E. Cristensen, R.D. Rabbitt, and M.I. Miller. Deformable templates using large deformation kinematics. IEEE Trans. Image Processing, 5: , [7] S. Clavet, P. Beaudoin, and P. Poulin. Particle-based viscoelastic fluid simulation. Eurograpics/ACM SIGGRAPH Symp. Computer Animation, To appear. [8] C. Davatzikos. Spatial transformation and registration of brain images using elastically deformable models. Computer Vision and Image Understanding, 66: , [9] L.D. Landau and E.M. Lifsitz. Fluid Mecanics. Pergamon,

11 [10] D. Ruecert. Nonrigid registration: tecniques and applications. Medical Image Registration, 13: , [11] B.C. Vemuri, J. Ye, Y. Cen, and C.M. Leonard. Image registration via level-set motion: applications to atlas-based segmentation. IEEE Trans. Medical Image Analysis, 7:1 20, [12] G. Wollny and F. Kruggel. Computational cost of nonrigid registration algoritms based on fluid dynamics. IEEE Trans. Medical Imaging, 21: ,