Solution-driven Adaptive Total Variation Regularization

Transcription

1 1/15 Solution-driven Adaptive Total Variation Regularization Frank Lenzen 1, Jan Lellmann 2, Florian Becker 1, Stefania Petra 1, Johannes Berger 1, Christoph Schnörr 1 1 Heidelberg Collaboratory for Image Processing (HCI) University of Heidelberg 2 Institute for Mathematics and Image Computing (MIC) University of Lübeck SIAM IS 16, Albuquerque, May 2016

2 Inverse Problems for Image Restoration 2/15 Variational approach with min u S(u, f) + αr(u) S : data term depending on data f R : regularization term α : regularization parameter Example: total variation R(u) = TV(u) = u(x) 2 dx

3 Types of Regularization 3/15 Adaptive Regularization penalization of u (or higher derivatives) varies locally data-driven: solution-driven: adaptivity depends on input data f or guidance image u 0 adaptivity is steered by the unknown u itself Anisotropic Regularization directionally dependent penalization of u (or higher derivatives) example: TV a (u) := u(x) 1 dx = x u(x) + y u(x) dx

4 4/15 Prototypes of Adaptive TV 1 weighted TV: R(u) := α(x, v) u(x) 2 dx

5 4/15 Prototypes of Adaptive TV 1 weighted TV: R(u) := α(x, v) u(x) 2 dx 2 anisotropic TV I: R(u) := with rotation matrix R(x, v) R(x, v) u(x) 1 dx

6 variable v: either f, a guidance image u 0 or u. 4/15 Prototypes of Adaptive TV 1 weighted TV: R(u) := α(x, v) u(x) 2 dx 2 anisotropic TV I: R(u) := with rotation matrix R(x, v) R(x, v) u(x) 1 dx 3 anisotropic TV II: R(u) := u(x) A(x, v) u(x) dx with symmetric positive-definite matrix A(x)

7 Related work I 5/15 1 Esedoglu & Osher, Decomposition of images by the anisotropic Rudin-Osher-Fatemi model, CPAA, Steidl & Teuber, Anisotropic smoothing using double orientations, SSVM, Grasmair, Locally Adaptive Total Variation Regularization, SSVM Dong et al., Automated Regularization Parameter Selection in Multi-Scale Total Variation Models for Image Restoration, IJCV, Bayram & Kamasak, Directional total variation, IEEE Signal Proc. Letters, 2012

8 6/15 Related work II 6 Lefkimmiatis, Roussos et al., Convex generalizations of total variation based on the structure tensor, 2015 R(u) = ( λ 1 (u), λ 2 (u)) p dx where λ 1 (u), λ 2 (u) are the eigenvalues of the structure tensor J(u). 7 Åström et al., A Tensor Variational Formulation of Gradient Energy Total Variation, 2015 u A(u) u R(u) = dx u 2 8 Estellers et al., Adaptive Regularization with the Structure Tensor, 2015 R(u) = R Jk (u) where J k is an iteratively updated estimate of the structure tensor.

9 General Definition 7/15 For u L 1 () { } R(u) := sup u div ϕ dx ϕ Cc 1 (; R n ), ϕ(x) D(x) with local constraint sets D(x): closed convex bounded non-empty (containing B c (0) for some c > 0) Higher order TV: exchange operator div.

10 Our approach 8/15 For some arbitrary image v define R v (u) := sup{ u div ϕ dx ϕ Cc 1 (; R n ), ϕ(x) D(x, v)}. Examples for D(x, v): R1: balls with radius α(x, v) depending on v 2. R2: ellipses oriented/scaled w.r.t. the eigenvectors/eigenvalues of structure tensor J(v). R1 R2

11 Our approach 9/15 Search for a fix point of v T (v) := arg min S(u, f) + R v (u). u (assuming a strictly convex problem).

12 Our approach 9/15 Search for a fix point of v T (v) := arg min S(u, f) + R v (u). u (assuming a strictly convex problem). After having found a fixed point u, we have u = arg min S(u, f) + R u (u). u solution-driven adaptive total variation regularization.

13 Theory 10/15 Under sufficient conditions Continuous case: existence of a fixed point for a sub-class of R v s, The proof uses the Himmelberg fixed point theorem. Discrete case: existence and uniqueness of a fixed point theory is via quasi-variational inequality problems: find p D(p ) h (f div h p ), p p 0 p D(p ) Details see: F. L., Adaptive Variational Regularization Techniques in Image Processing and Computer Vision. Habilitation Thesis, University of Lübeck, 2015

14 Results Denoising noisy image R1 data-driven R1 solution-driven SSIM 1 index: standard TV R2 data-driven R2 solution-driven (sub-optimal parameters) 1 Wang et al. Image quality assessment: from error visibility to structural similarity., /15

15 Results Deblurring blurry image R1 data-driven R1 solution-driven SSIM 2 index: standard TV R2 data-driven R2 solution-driven (optimal parameters) 2 Wang et al. Image quality assessment: from error visibility to structural similarity., /15

16 Results Inpainting blurry image standard TV SSIM index: R2 data-driven R2 solution-driven (optimal parameters) 13/15

17 Conclusion 14/15 Increasing use of adaptive regularization techniques. Solution-driven adaptivity is preferable. Theoretical issues in general: function spaces, (non)-convexity, existence & uniqueness Advantage of our approach: inner problem is convex and well-posed & theory on fixed points.

18 15/15 Solution-driven Adaptive Total Variation Regularization Frank Lenzen 1, Jan Lellmann 2, Florian Becker 1, Stefania Petra 1, Johannes Berger 1, Christoph Schnörr 1 1 Heidelberg Collaboratory for Image Processing (HCI) University of Heidelberg 2 Institute for Mathematics and Image Computing (MIC) University of Lübeck SIAM IS 16, Albuquerque, May 2016

19 Himmelberg Fixed Point Theorem 1/2 Theorem (Theorem 2.3 in Agarwal and O Regan, 2002) Let E be a Banach space and let C be a closed convex subset of E. Then any weakly compact, weakly sequentially upper-semicontinuous map F : C K(C) has a fixed point. Proof utilizes Theorem (Himmelberg Fixed Point Theorem) Let T be a nonvoid convex subset of a separated locally convex space L. Let F : T T be a u.s.c. multimap such that F (x) is closed and convex for all x T, and F (T ) is contained in some compact subset C of T. Then F has a fixed point.

20 Results A posteriori distribution 2/2