Inter and Intra-Modal Deformable Registration:

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1 Inter and Intra-Modal Deformable Registration: Continuous Deformations Meet Efficient Optimal Linear Programming Ben Glocker 1,2, Nikos Komodakis 1,3, Nikos Paragios 1, Georgios Tziritas 3, Nassir Navab 2 4 July GALEN group laboratoire de mathématiques appliquées aux systèmes ecole centrale paris 2 computer aided medical procedures & augmented reality technische universität münchen 3 computer science department university of crete

2 Outline Introduction & Motivation Image Registration based on Discrete Labeling Optimization using Linear Programming Results & Conclusions In this presentation everything is intensity-based camp+ar department of computer science technische universität münchen 11 July

3 Introduction Source and target image f : Ω R g : Ω R Ω R d with d {2, 3} Image relation g(x) = h f(t (x)) T : Ω Ω h : R R non-linear transformation non-linear relation on intensities camp+ar department of computer science technische universität münchen 11 July

4 Registration as an Optimization Problem Energy formulation Z E(T ) = ρ h (g(x), f(t (x))dx min! Ω ρ h : R R R distance measure The aim of registration is to recover the transformation which involves the definition of a transformation type the definition of a distance/similarity measure the definition of an optimization procedure camp+ar department of computer science technische universität münchen 11 July

5 Review of Registration Methods Types of transformations Rigid, affine, projective Basis functions, Spline-based Finite Element Models, Distance/Similarity measures SAD, SSD, NCC, NMI, CR, Optimization methods Variational Gradient-based Direct search (Simplex, Powell-Brent, Best Neighbor) camp+ar department of computer science technische universität münchen 11 July

6 Motivation What is expected from an optimal registration method? Independent from the choice of the transformation type Independent from the choice of the distance/similarity measure Guarantee of a globally optimal solution Reasonable computational complexity camp+ar department of computer science technische universität münchen 11 July

7 Our Contributions Novel deformable registration framework based on discrete labeling and linear programming Our framework bridges the gap between continuous deformations and discrete optimization Gradient-free and flexible in the choice of the distance measure Guaranteed optimality properties on the solution Computational efficient and tractable camp+ar department of computer science technische universität münchen 11 July

8 Image Registration based on Discrete Labeling camp+ar department of computer science technische universität münchen 11 July

9 Local Registration Deformation grid providing a continuous and dense deformation field T (x) = x + D(x) with D(x) = X p G η( x p ) d p In our implementation we use Free Form Deformation D(x) = 3X 3X B l (u)b m (v) d i+l,j+m l=0 m=0 [Rueckert99, Schnabel01, Rohlfing03, ] camp+ar department of computer science technische universität münchen 11 July

10 Energy Formulation Reformulation of the optimization problem E data (T ) = X Z η 1 ( x p ) ρ h (g(x), f(t (x)))dx p G Ω Smoothness term E smooth (T ) = X p G φ( G d p ) Registration task E total = E data + E smooth min! camp+ar department of computer science technische universität münchen 11 July

11 Discrete Optimization Problem V pq (u p, u q ) General energy formulation V p (u p ) p q r s t u E total = E data + E smooth min! v w x Markov Random Field (MRF) formulation for discrete labelings E total (u) = X V p (u p ) + X V pq (u p, u q ) p G p,q E Data term = singleton potentials Smoothness term = pairwise potentials camp+ar department of computer science technische universität münchen 11 July

12 Discretization of Parameter Space Set of labels and a discretized deformation space L = {u 1,..., u i } Θ = {d 1,..., d i } +Y Max Steps +Y -X +X -X +X -Y Sparse sampling -Y Dense sampling camp+ar department of computer science technische universität münchen 11 July

13 Data Term MRF singleton potentials: L G cost matrix E data (u) = X Z η 1 ( x p )ρ h (g(x), f(t (x)))dx p G Ω X p G V p (u p ) Problem: singleton potentials are not independent! camp+ar department of computer science technische universität münchen 11 July

14 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=0 y=0 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

15 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=10 y=0 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

16 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=10 y=-10 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

17 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=0 y=-10 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

18 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=-10 y=-10 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

19 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=-10 y=0 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

20 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=-10 y=10 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

21 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=0 y=10 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

22 Fast Approximation of Singleton Potentials Approximation of label costs simultaneously for all nodes Nodes Labels Current label x=10 y=10 Single potential look-up table camp+ar department of computer science technische universität münchen 11 July

23 Smoothness Term MRF pairwise potentials: L L cost matrix E smooth (u) = X V pq (u p, u q ) p,q E e.g. truncated absolute difference (piecewise smooth) V pq (u p, u q ) = λ pq min ( d u p d u q, T ) Note: smoothness function can vary locally camp+ar department of computer science technische universität münchen 11 July

24 MRF Formulation of Image Registration E total (u) = X p G V p (u p ) + X p,q E V pq (u p, u q ) Opens the door to MRF optimization techniques camp+ar department of computer science technische universität münchen 11 July

25 Optimization using Linear Programming camp+ar department of computer science technische universität münchen 11 July

26 Primal-Dual Schema Say we seek an optimal solution x* to the following integer program (this is our primal problem): (NP-hard problem) To find an approximate solution, we first relax the integrality constraints to get a primal & a dual linear program: primal LP: dual LP: camp+ar department of computer science technische universität münchen 11 July

27 Primal-Dual Schema Goal: find integral-primal solution x, feasible dual solution y such that their primal-dual costs are close enough, e.g., T cx T by f * T cx T * cx f * dual dual cost cost of of solution y T by T * cx cost cost of of optimal integral solution x* x* T cx primal primal cost cost of of solution x Then x is an f * -approximation to optimal solution x* camp+ar department of computer science technische universität münchen 11 July

28 Primal-Dual Schema The primal-dual schema works iteratively sequence of dual costs T k cx T k by f * sequence of primal costs T 1 by T 2 b y T k b y T * cx T k cx cx T 2 unknown optimum T 1 cx camp+ar department of computer science technische universität münchen 11 July

29 Primal-Dual Schema for MRFs (only one label assigned per vertex) enforce consistency between variables x p,a, x q,b and variable x pq,ab Binary variables x p,a p,a =1 x pq,ab pq,ab =1 label a is is assigned to to node p labels a, a, b are assigned to to nodes p, p, q camp+ar department of computer science technische universität münchen 11 July

30 Primal-Dual Schema for MRFs During the PD schema for MRFs, it turns out that each update of of primal and dual variables solving max-flow in in appropriately constructed graph Resulting flows tell us how to update both: the dual variables, as well as the primal variables for each iteration of primaldual schema Fast-PD camp+ar department of computer science technische universität münchen 11 July

31 Fast-PD MRF optimization method based on duality theory of Linear Programming (the Primal-Dual schema) Can handle a very wide class of MRFs Can guarantee approximately optimal solutions (worst-case theoretical guarantees) Can provide tight certificates of optimality per-instance (per-instance guarantees) Provides significant speed-up for static and dynamic MRFs Komodakis, N., Tziritas, G., Paragios, N. Fast, Approximately Optimal Solutions for Single and Dynamic MRFs. Computer Vision and Pattern Recognition 2007 camp+ar department of computer science technische universität münchen 11 July

32 MRF Hardness local optimum global optimum approximation MRF hardness exact global optimum MRF pairwise potential linear metric arbitrary Fast-PD camp+ar department of computer science technische universität münchen 11 July

33 Registration Algorithm for i=1:no_iterations 1: setup_label_sets() 2: precompute_single_potential_matrix() 3: precompute_pairwise_potential_matrix() 4: compute_discrete_labeling() 5: update_deformation() end Gaussian image pyramids & multi-level deformation grids are used for a hierarchical registration approach camp+ar department of computer science technische universität münchen 11 July

34 Results & Conclusions camp+ar department of computer science technische universität münchen 11 July

35 Comparison Schnabel01 Our Method Our Method SSD error Running time Before registration > 2 hours < 2 min < 2 min camp+ar department of computer science technische universität münchen 11 July

36 Visual Results camp+ar department of computer science technische universität münchen 11 July

37 Validation of Distance Measures camp+ar department of computer science technische universität münchen 11 July

38 Validation of Distance Measures camp+ar department of computer science technische universität münchen 11 July

39 Results (MICCAI 2007) Mean Intensity Image Variance Intensity Image Segmentation ρ Atlas (I μ, I σ 2, I new ) = 1 Ω P x Ω (I μ (x) I new (x)) 2 2I σ 2 (x) Method DSC Sensitivity Specificity Interaction Cartilage Grau et al (0.01) % % 5-10 min Tibia, Femur, Patella Dam et al (n/a) % % Max 10 min Tibia, Femur Cheong et al (0.15) % n/a 0 Medial Tibia Cheomg et al (0.09) % n/a 0 Lateral Tibia Folkesson et al (0.03) % % 0 Tibia, Femur Our Approach 0.83 (0.06) % % 0 Patella camp+ar department of computer science technische universität münchen 11 July

40 Conclusions What is provided by our registration framework? Independent from the choice of the transformation type? Grid-based but independent from weighting function. Independent from the choice of the distance/similarity measure? Yes. We are gradient-free. Guarantee of a globally optimal solution? Quasi-yes (from an optimization point of view). Reasonable computational complexity? Yes (256x256 in 2 seconds, 256x192x64 in 1 minute). camp+ar department of computer science technische universität münchen 11 July

41 Latest Work & Future Directions On-the-fly estimation of locally varying deformation spaces (done!) Incorporate global registration & different deformation models Introducing domain knowledge such as priors on the deformation space Belief propagation networks GPU implementation camp+ar department of computer science technische universität münchen 11 July

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