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
Dense Image Registration through MRFs and Efficient Linear Programming
Revised Manuscript Dense Image Registration through MRFs and Efficient Linear Programming Ben Glocker a,b,, Nikos Komodakis a,c, Georgios Tziritas c, Nassir Navab b, Nikos Paragios a a GALEN Group, Laboratoire
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