Random walks for deformable image registration Dana Cobzas

Transcription

1 Random walks for deformable image registration Dana Cobzas with Abhishek Sen, Martin Jagersand, Neil Birkbeck, Karteek Popuri Computing Science, University of Alberta, Edmonton

2 Deformable image registration?t I target J source Outline: Continuous formulation of deformable registration Discrete formulation and related work Diffusion and regularization from continuous to discrete Random walker for deformable registration Examples and discussion

3 Deformable image registration?t transformation I target I,J :R, R d T : R d, T =Idu J source Eu= E D I, J T u E R u Data similarity regularization u=argmin u E u u=(u x,u y ) displacement field

4 Variational image registration u=argmin u x I x, J xu x d x Data similarity (point-wise score) x J x, u x d x Adaptable regularization + EXAMPLE: SSD Linear isotropic regularization I x, J xux=i x J xu x 2 J x, u x=w J x 2 u x 2 u y 2 scalar valued diffusivity

5 Euler-Lagrange equations min u E D I, J T x w. u x 2 u y 2 d x E D u x w. u x =0 E D u y w. u y =0 Linear isotropic diffusion Continuous solution to deformable registration: Nonlinear optimization (gradient descent, Gauss-Newton...) Semi-implicit scheme to solve the EL PDEs Regularization can be decoupled from the data term update demon's algorithm Main limitation local minima > need for better frameworks

6 Discrete deformable registration u i D I I i, i=1..n J i, i=1..n J Image = Graph G = (N, E) d k Quantized deformations D = {d 1, d 2,...,d K } Labels L = {u 1, u 2,...,u K } Note: D can be very large Ex: deformations [0-20pix] 41 3 = labels

7 Discrete deformable registration u i D I I i, i=1..n J i, i=1..n J Image = Graph G = (N, E) Registration as labeling : MRF Quantized deformations D = {d 1, d 2,...,d k } Labels L = {u 1, u 2,...,u k } Note: D can be very large Ex: deformations [0-20pix] 41 3 = labels Eu= i N i u i i, j E ij u i, u j data similarity Regularization Interaction

8 Related work Eu= i N i u i i, j E ij u i, u j Global optimum ij metric graph cut optimization [T.W. Tang et al. MICCAI 2007][L. Tang et al. MICCAI 2010] More complex interaction terms linear programming method based on primal-dual principle [Glocker et al. MIA 2008]

9 Related work Eu= i N i u i i, j E ij u i, u j Global optimum ij metric graph cut optimization [T.W. Tang et al. MICCAI 2007][L. Tang et al. MICCAI 2010] More complex interaction terms linear programming method based on primal-dual principle [Glocker et al. MIA 2008] We propose : Gaussian MRF for deformable registration solved using random walker algorithm [Grady PAMI 2006, CVPR 2005] Fast algorithm Can easily incorporate a large number of displacements Robustness to noise Unique global minimum (has to be relaxed)

10 Related work Eu= i N i u i i, j E ij u i, u j ij metric graph cut optimization [T.W. Tang et al. MICCAI 2007][L. Tang et al. MICCAI 2010] More complex interaction terms linear programming method based on primal-dual principle [Glocker et al. MIA 2008] Here: Gaussian MRF for deformable registration solved using random walker algorithm [Grady PAMI 2006, CVPR 2005] Fast algorithm Can easily incorporate a large number of displacements Robustness to noise Unique global minimum (has to be relaxed) Regularization E R u= u 2 dxdy diffusion u=0 NEXT: - diffusion and regularization - diffusion on graphs - RW for image registration - Experiments and discussion

11 Diffusion u t Uniform =div u=u u t Isotropic =div w u u0=u 0 u0=u 0 u(0) w s 2 = 1 s 2 s= u 0 u(t) u(t)... Scalar-valued diffusivity Changes the metric of the space Slows down/blocks diffusion at edges Steady state

12 Diffusion u t Uniform =div u=u u t Isotropic =div w u u0=u 0 u0=u 0 u(0) w s 2 = 1 s 2 s= u 0 u(t) u(t)... Steady state Scalar-valued diffusivity Changes the metric of the space Slows down/blocks diffusion at edges Numerical solution: Parabolic PDEs fully implicit time discretization > elliptic PDEs solve with implicit or semi-implicit FD schemes

13 Regularization and diffusion f u? Energy functional : Euler Lagrange Equation : E u u= 1 2 f u 2 u 2 dxdy u=argmin u E u u u f =u data regularization

14 Regularization and diffusion f u? Energy functional : Euler Lagrange Equation : data E u u= 1 2 f u 2 u 2 dxdy u=argmin u E u u u f regularization Uniform diffusion u =u t =u Fully implicit time discretization of the uniform diffusion with initial value f and time step α

15 Regularization Uniform Isotropic w E R u= u 2 dxdy E R u= w f 2 u 2 dxdy Euler Lagrange u f =u u f = w. u

16 Regularization Uniform Isotropic w E R u= u 2 dxdy E R u= w f 2 u 2 dxdy Euler Lagrange u f =u u f = w. u Deformable registration min u E D I, J T x w. u x 2 u y 2 d x E D u x w. u x =0 E D u y w. u y =0

17 u : u=0 ux=f B, x K B Discrete diffusion Finite differences + implicit scheme Boundary conditions u i-1,j-1 u i-1,j u i-1,j+1 u i,j-1 u i,j u i,j+1 u i+1,j-1 u i+1,j u i+1,j+1 Pixel location u Nx1 L NxN Lu=0 [ L u B B][ u B] u B L f =0 L u u u = B T f B

18 u: u=0 ux=f B, x K B Discrete diffusion Finite differences + implicit scheme Boundary conditions u i-1,j-1 u i-1,j u i-1,j+1 u i,j-1 u i,j u i,j+1 u i+1,j-1 u i+1,j u i+1,j+1 Pixel location Lu=0 [ L u B B][ u B] u B L f =0 L u u u = B T f B Generalize L isotropic diffusion Combinatorial Laplacian L ij = d i = k w ik w ij if i= j if i, j E 0 otherwise Positive semi-definite (PSD)

19 u=0 ux=f B, x K B Discrete diffusion Boundary conditions Combinatorial Dirichlet integral u: Finite differences + implicit scheme 1 u i-1,j-1 u i-1,j u i-1,j u i,j-1 u i,j u i,j+1 1 u i+1,j-1 u i+1,j u i+1,j+1 Pixel location D [u]= u 2 dx D[u ]=u t A T A u=u t L u A (ij)k incidence matrix (combinatorial gradient) Critical points = minima [L-PSD] Lx=0 Lu=0 [ L u B B][ u B] u B L f =0 L u u u = B T f B Generalize L isotropic diffusion Combinatorial Laplacian L ij = d i = k w ik w ij if i= j if i, j E 0 otherwise Positive semi-definite (PSD)

20 Random walker algorithm Lu=0 [ L u B B][ u B] u B L f =0 L u u u = B T f B [Grady PAMI 2006] Combinatorial Laplacian L ij = d i = k w ik if i= j if i, j E 0 otherwise w ij w ij =exp I i I j 2 f B seeds ; u i prob that a random walker from node i first reaches a seed.

21 RW for deformable registration u i D Quantized deformations D = {d 1, d 2,...,d K } Labels L = {u 1, u 2,...,u K } I i, i=1..n Registration as labeling : MRF J i, i=1..n E u= i N i u i i, j E ij u i,u j data similarity potential Regularization Interaction

22 RW for deformable registration u i D Quantized deformations D = {d 1, d 2,...,d K } Labels L = {u 1, u 2,...,u K } I i, i=1..n Registration as labeling : MRF J i, i=1..n E u= i N i u i i, j E ij u i,u j data similarity potential Gaussian MRF : Regularization Interaction E k u k = i N l=1: K,l k i l u i k 2 i k 1 u i k 2 ij E w ij u i k u j k 2 Modifications : 1. u i k = prob of node i having label u k 2. Gaussian MRF with interaction u i k,u j k =w ij u i k u j k 2 3. 'priors' for probability of a displacement i k =exp I i J id k 2

23 Gaussian MRF for deformable registration Displacement probability i k =exp I i J id k 2 1 if displacement d k fits perfectly the similarity score Small - otherwise Data potential i u i k = l =1: K,l k i l u i k 2 i k 1 u i k 2 Displacement l k has high prob. > low prob. for label k Displacement k has high prob. > high prob. for label k

24 Gaussian MRF for deformable registration Displacement probability i k =exp I i J id k 2 1 if displacement d k fits perfectly the similarity score Small - otherwise Data potential Regularization Interaction i u i k = l =1: K,l k i l u i k 2 i k 1 u i k 2 Displacement l k has high prob. > low prob. for label k u i k,u j k =w ij u i k u j k 2 w ij =exp J i J j 2 Displacement k has high prob. > high prob. for label k Recall : regularization term (Dirichlet integral) Encourages two neighbors to have similar labels if the image intensity is similar. D[u]= w J 2 u 2 dx D[ u]=u t Lu

25 RW with priors [following Grady CVPR 2005] E k u k = i N l=1 : K, l k i l u i k 2 i k 1 u i k 2 ij E w ij u i k u j k 2 Combinatorial Laplacian L L ij = d i = k w ik if i= j if i, j E 0 otherwise w ij k =diag k

26 RW with priors [following Grady CVPR 2005] E k u k = i N l=1: K,l k i l u i k 2 i k 1 u i k 2 ij E w ij u i k u j k 2 Combinatorial Laplacian L L ij = d i = k w ik if i= j if i, j E 0 otherwise w ij k =diag k E k u k = l=1 : K,l k u kt l u k 1 u k T k 1 u k u kt Lu k Minimum when K L l=1 l u k = k Note: The combined matrix still positive semi-definite One equation system per label!!

27 Augmented graph K L l=1 l u k = k Combinatorial Laplace equation for an augmented graph w ij =exp J i J j 2 Large (1) if neighbors have similar color (allows diffusion)

28 Augmented graph K L l=1 l u k = k Combinatorial Laplace equation for an augmented graph u u 1l u K i l =exp I i J id l 2 w ij =exp J i J j 2 Large (1) if Matching color (SSD) Large (1) if neighbors have similar color (allows diffusion)

29 Implementation Computational cost Need to solve for each displacement label (actually K-1) Size of K L l=1 l u k = k Multi-resolution pyramid Loose global optimality displacements # range ¼ 60 [15,15] ½ 40 [-10,10] 1 30 [-7,7]

30 Implementation Computational cost Need to solve for each displacement label (actually K-1) Size of K L l=1 l u k = k Multi-resolution pyramid Loose global optimality displacements # range ¼ 60 [15,15] ½ 40 [-10,10] 1 30 [-7,7]

31 Synthetic image + Experiments: Quality of deformations known deformation field Demon's alg. RW 256x256 image 3 levels resol. 160sec (MATLAB) Ang. err. (deg) 1.58 ± ± 0.97 Mag. Err. (deg) 1.12 ± ± 2.89 SSD [0-255]

32 Brain MRI with WM/GM/CSF segmentation source target Reg. Demons SSD=8.82 Reg. RW SSD=3.38 Data from Brain segmentation Repository Dice

33 Dice Abdominal CT with muscle segmentation source target Data from CCI, Edmonton Reg. Demons SSD=15.46 Reg. RW SSD=9.64

34 Discussion CONTRIBUTION: New discrete formulation of deformable image registration based on the RW method Image dependent regularization term Global solution with respect to discretization LIMITATIONS AND EXTENSIONS : Computational cost : Solve a large equation system for each label FFD Requirement of a point-wise similarity score Eu= i N i u i i, j E ij u i, u j Possible solution Lower dim models : Compute displacements only at control points Approximate non-local scores in a neighborhood defined by the control points FFD