Variational methods for restoration of phase or orientation data

Transcription

1 Variational methods for restoration of phase or orientation data Martin Storath joint works with Laurent Demaret, Michael Unser, Andreas Weinmann Image Analysis and Learning Group Universität Heidelberg Symposium on Mathematical Optics, Image Modelling, and Algorithms June 21, 2016

2 Phase or orientation data Phase or orientation data: images (or time series) whose pixels take their values on the unit circle i.e. f ij S 1 = T π π/2 Direction [rad] 0 -π/ Time [h] Hourly wind directions at station VENF1 (Venice, FL) in 2014 Interferometric SAR image of Vesuvius used for creation of digital elevation maps Goal: devise variational regularization (total variation, Potts, Mumford-Shah,...) for cirlce-valued data 2 / 20

3 Total variation regularization for circle-valued data Total variation (TV) regularization for real valued data (Rudin/Osher/Fatemi 92) u = arg min u R M N γ u ij u i+1,j + γ u ij u +1 + u ij f ij p f R M N given image, γ > 0 model parameter widely used for edge/jump preserving regularization convex computationally tractable 3 / 20

4 Total variation regularization for circle-valued data Total variation (TV) regularization for real valued data (Rudin/Osher/Fatemi 92) u = arg min u R M N γ u ij u i+1,j + γ u ij u +1 + u ij f ij p f R M N given image, γ > 0 model parameter widely used for edge/jump preserving regularization convex computationally tractable TV for circle-valued data, f T M N (Giaquinta/Modica/Soucek 93; Strekalovskiy/Cremers 11) u = arg min u T M N γ d T (u ij, u i+1,j ) + γ d T (u ij, u +1 ) + d T (u ij, f ij ) p where d T (x, y) is the arc length distance between two point x, y S 1 nonlinear data space vector space methods not applicable nonconvex globally optimal solutions possible? 3 / 20

5 L 1 -TV for circle-valued data in 1D L 1 -TV for circle-valued signals f T N, (e.g., time series of orientations), arg min u T N is computationally tractable: N d T (u n, u n+1 ) + d T (u n, f n ), N 1 γ n=1 n=1 Theorem (S., Weinmann, Unser) There is an exact algorithm for the total variation problem with circle-valued data. Its complexity is O(KN), where K N the number of unique values in f. Sketch of proof: Show that there is a minimizer whose values are subset of the data values and its antipodal points. Utilize the Viterbi algorithm (a type of dynamic programming). Generalize infimal convolution for fast message passing (Felzenszwalb/Huttenlocher) to circular data. S., Weinmann, Unser. Exact algorithms for L1-TV regularization of real-valued or circle-valued signals. SIAM J. Sci. Comp. (2016). 4 / 20

6 Discontinuity-preserving smoothing of time series π π/2 Direction [rad] 0 -π/ Time [h] Hourly wind directions at station VENF1 (Venice, FL) in 2014 π/2 Direction [rad] 0 -π/ Time [h] Global minimizer of L 1 -TV with circle-valued data (CPU time 3.3 sec) Data available at 5 / 20

7 TV for circle-valued data NP hard in 2D TV problem for circle-valued images f ij T = S 1, arg min γ d T (u, u +1 ) + γ d T (u i+1,j, u ) + d T (u ij, f ij ), u T m n NP-hard in 2D (Cremers/Strekalovskiy 13) resort to approximative strategies 6 / 20

8 TV for circle-valued data NP hard in 2D TV problem for circle-valued images f ij T = S 1, arg min γ d T (u, u +1 ) + γ d T (u i+1,j, u ) + d T (u ij, f ij ), u T m n NP-hard in 2D (Cremers/Strekalovskiy 13) resort to approximative strategies Our approach: Cyclic proximal point algorithm (CPPA) devise algorithm for general (Riemannian) manifold-valued TV problem arg min γ d M (u, u +1 ) + γ d M (u i+1,j, u ) + d M (u ij, f ij ), u M m n where d M is the distance induced by the Riemannian metric algorithm should be globally convergent on nice manifolds M, e.g., tensor manifolds 6 / 20

9 Tensor manifolds in diffusion tensor imaging Diffusion tensor imaging (DTI) (Basser/Mattiello/LeBihan 94) Based on gradient weighted MR-images D v w.r.t. direction v R 3 Relation to diffusion tensor P by Stejskal-Tanner equation D v = A 0 e b vt Pv Clinical application: study of neurodegenerative diseases, e.g., Alzheimer The diffusion tensor manifold (Pennec et al. 04) A diffusion tensor P is a symmetric positive definite 3 3 matrix (P Pos 3 ). Pos 3 is a Riemannian manifold with the Riemannian metric g P (A, B) = trace(p 1/2 AP 1 BP 1/2 ), where P Pos 3, and A, B tangent vectors (symmetric 3 3 matrices) at P. affine invariant distance, unique geodesics A diffusion tensor image visualized by ellipsoids 7 / 20

10 Proposed minimization approach Split the TV functional into atomic functionals F(u) = γ d(u i, u i 1 ) + d(u j, f j ) p = F i (u) + G(u), i j i where F i (u) = γ d(u i, u i 1 ) and G = j d(u j, f j ) p. Use a cyclic proximal point strategy (Bertsekas, Bacak), i.e., iterate the proximal mappings of G and F i, 1 prox λg (u) = arg min d(u, v 2 v)2 + λg(v), 1 prox λfi (u) = arg min d(u, v 2 v)2 + λf i (v). (For simplicity: 1D functional; 2D/3D analogously) Weinmann, Demaret, S. Total variation regularization of manifold-valued images. SIAM J. Imag. Sci. (2014) 8 / 20

11 Computation of proximal points on a manifold Key observation: Proximal mappings can be computed explicitly Proximal mapping of G given by 2λ (1+2λ) prox λg (u) i = [u i, f i ] t, t = d(u i, f i ), for p = 2, min(λ, d(u i, f i )), for p = 1. Proximal mapping of F i given by u j, if j i, i 1, prox λfi (u) j = [u i, u i 1 ] t, if j = i, [u i 1, u i ] t, if j = i 1, with t = min(λγ, 1 2 d(u i, u i 1 )). Efficient computation of geodesics [P, Q] and Riemannian distances d(p, Q) [P, Q] t = exp P (t log P (Q)) and d(p, Q) = log P (Q) P via Riemannian exponential and logarithmic mappings (Pennec et al. 06) log P Q = P 1 2 log(p 1 2 QP 1 2 )P 1 2, and exp P A = P 1 2 exp(p 1 2 AP 1 2 )P 1 2. Weinmann, Demaret, S. Total variation regularization of manifold-valued images. SIAM J. Imag. Sci. (2014) 9 / 20

12 Results of TV regularization for synthetic DT images Original Rician noise (σ = 90) Result using the proposed method 10 / 20

13 Results for real DT images Diffusion tensor image of a human brain TV result using proposed cyclic proximal point algorithm (CPU-Time: 496 sec) Data by courtesy of the CAMINO project 11 / 20

14 Analytic results Cartan-Hadamard manifolds: (e.g., DTI manifold Pos 3 ) For each x 0, x 1 there is a point y on the geodesic [x 0, x 1 ] such that for every z, d 2 (z, y) 1 2 d2 (z, x 0 ) d2 (z, x 1 ) 1 4 d2 (x 0, x 1 ). Theorem (Weinmann/Demaret/S. 14) In a Cartan-Hadamard manifold (complete, simply connected), the proposed cyclic proximal point algorithm for TV minimization converges towards a global minimizer. 12 / 20

15 Analytic results Cartan-Hadamard manifolds: (e.g., DTI manifold Pos 3 ) For each x 0, x 1 there is a point y on the geodesic [x 0, x 1 ] such that for every z, d 2 (z, y) 1 2 d2 (z, x 0 ) d2 (z, x 1 ) 1 4 d2 (x 0, x 1 ). Theorem (Weinmann/Demaret/S. 14) In a Cartan-Hadamard manifold (complete, simply connected), the proposed cyclic proximal point algorithm for TV minimization converges towards a global minimizer. Related approaches for TV with manifold-valued data a priori discretization of manifold and convex relaxation (Lellmann et al. 13) iteratively reweighted least squares minimization (Grohs/Sprecher 15) Douglas-Rachford splitting for manifolds with constant sectional curvature (Bergmann et al. 16) 12 / 20

16 CPPA for phase data Application of proposed CPPA to circle-data using with a = e iθ and v ] π; π[, and exp a (v) = e i(θ+v), exp 1 a (b) = arg(b/a), where a, b complex number representations of values on the unit circle Results Interferometric SAR image of Vesuvius (real data) L 2 -TV L 1 -TV TV with Huber data term S 1 -values visualized as hue component in the HSV colorspace. 13 / 20

17 Further applications of our methods for manifold-valued regularization Shape data (RP d manifold) Top: Time series of segmentation curves; Bottom: TV regularization in shape space X-ray tensor tomography (Malecki et al. 14) (Pos3 manifold) Carbon fibers X-ray tensors TV denoising Position data (SO (3) manifold) Original Noisy SO(3) data 100 TV denoising Baust, Demaret, S., Navab, Weinmann. Total variation regularization for shape signals. CVPR (2015) M. Wieczorek et al. Total variation regularization for x-ray tensor tomography. Fully3D (2015) Weinmann, Demaret, S. Total variation regularization of manifold-valued images. SIAM J. Imag. Sci. (2014) 14 / 20

18 Mumford-Shah regularization for vectorial data Mumford-Shah model for vectorial data (Mumford/Shah 85/ 89; Blake/Zisserman 87) arg min γ length(k ) + α u,k Z Z u 2 dx + (u f )p dx KC where K R2 denotes a discontinuity set { piecewise smooth approximation + Preserves discontinuities Computationally challenging Blurred and noisy image Tikhonov regularization Mumford-Shah Induced edge set K 15 / 20

19 Mumford-Shah regularization for manifold-valued data Mumford-Shah model for manifold-valued data arg min γ K + α Du(x) q dx + 1 u,k q p Ω\K Ω d(u(x), f(x)) p dx Prior work: Level-set active contour approach for two-phase variant (manifold-valued Chan-Vese model) (Wang/Vemuri 04) Weinmann, Demaret, S. Mumford-Shah and Potts regularization for manifold-valued data. J. Math. Imag. Vis. (2016) 16 / 20

20 Mumford-Shah regularization for manifold-valued data Mumford-Shah model for manifold-valued data arg min γ K + α Du(x) q dx + 1 u,k q p Ω\K Ω d(u(x), f(x)) p dx Prior work: Level-set active contour approach for two-phase variant (manifold-valued Chan-Vese model) (Wang/Vemuri 04) Our approach: (i) Finite difference discretization of Blake-Zisserman-type with the directional penalty 1 arg min x M m n p dp (x, f) + α R ω s Ψ as (x), Ψ a (x) = ψ(x ()+a, x ij ) and ψ(w, z) = 1 q min(γq/α, d(w, z)q ). s=1 Weinmann, Demaret, S. Mumford-Shah and Potts regularization for manifold-valued data. J. Math. Imag. Vis. (2016) 16 / 20

21 Algorithm for 2D Mumford-Shah problem (ii) Penalty decomposition arg min x 1,...,x R R ω s prαψ as (x s ) + d p (x s, f) + µ k d p (x s, x s+1 ). s=1 with an increasing coupling parameter µ k (iii) Block coordinate descent x k+1 arg min prω 1 1 αψ a1 (x) + d p (x, f) + µ k d p (x, x k R ), x x k+1 arg min prω 2 2 αψ a2 (x) + d p (x, f) + µ k d p (x, x k+1 ), x 1. x k+1 R arg min x prω R αψ ar (x) + d p (x, f) + µ k d p (x, x k+1 R 1 ). univariate Mumford-Shah-type problems 17 / 20

22 Exact solver for univariate Mumford-Shah problems (iv) Solve univariate Mumford-Shah problems 1 arg min x p i=1 where J is the jump set of x. n d(x i, f i ) p + α d(x i, x i+1 ) q + γ J(x), q i J(x) Theorem (Weinmann/Demaret/S. 16) In a Cartan-Hadamard manifold, there is an algorithm that produces a global minimizer for the univariate Mumford-Shah problem. Sketch of proof: Employ dynamic programming strategy (Friedrich et al. 08) subproblems of TV q -L p type Solve subproblems using CPPA. 18 / 20

23 Results for diffusion tensor images Result using proposed splitting method Corpus callosum of human brain (CAMINO project) Mumford-Shah regularization using proposed method 19 / 20

24 Summary Algorithm for TV regularization with circle-valued data in 1D Exact solver for univariate L 1 -TV Worst case complexity O(N 2 ) Algorithms for TV regularization for manifold-valued images Cyclic proximal point strategy with explicit computation of proxies Globally convergent algorithm for Cartan-Hadamard manifolds Satisfactory results for (non-convex) circle-valued data Algorithms for Mumford-Shah regularization with manifold-valued data Based on penalty decomposition, dynamic programming, and CPPA No restrictions on discontinuity curve (e.g. number of segments) 20 / 20

25 Thank you!