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Error Estimation in TV Imaging

Error Estimation in TV Imaging. Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münster. Joint Work with. Stan Osher (UCLA)

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Error Estimation in TV Imaging

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  1. Error Estimation in TV Imaging Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münster

  2. Joint Work with • Stan Osher (UCLA) mb-Osher, Inverse Problems 04 • Elena Resmerita, Lin He (Linz) mb-Resmerita-He, Computing 07 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  3. TV Imaging • Total variation methods are one of the most popular techniques in modern imaging • Basic idea is to model image, resp. their main structure (cartoon) as functions of bounded variation • Reconstructions seek images of as small total variation as possible Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  4. TV Imaging Z Z j j j j j j ( ) d d r r ¢ u u s u p u u x g x = = B V B V k k C · 1 1 2 g g 1 0 ; • Total variation is a convex, but not differentiable and not strictly convex functional „ “ • Banach space BV consisting of all L1 functions of bounded variation Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  5. Denoising Models r u ( ) f 0 t ( ) j j ¸ Z @ @ u r = = ¡ 2 ¢ u u = T V t 2 j j ( ) j j r f i ¡ + u u u m n ! T V 2 B V 2 u • ROF model Rudin-Osher Fatemi 89/92, Chambolle-Lions 96, Scherzer-Dobson 96, Meyer 01,… • TV flow Caselles et al 99-06, Feng-Prohl 03, .. Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  6. ROF Model j j ¸ ¸ f @ + 2 p u p u = T V ( ) f j ¤ @ 8 J X X ; 2 2 u p v : = ( ) h i ( ) g J J · + ¡ u p v u v ; • Optimality condition for ROF denoising • Dual variable p enters in ROF and TV flow – related to mean curvature of edges for total variation • Subdifferential of convex functional Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  7. ROF Model Reconstruction (code by Jinjun Xu) clean noisy ROF Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  8. ROF Model • ROF model denoises cartoon images resp. computes the cartoon of an arbitrary image, natural spatial multi-scale decomposition by varying l Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  9. Error Estimation ? • First question for error estimation: estimate difference of u (minimizer of ROF) and f in terms of l • Estimate in the L2norm is standard, but does not yield information about edges • Estimate in the BV-norm too ambitious: even arbitrarily small difference in edge location can yield BV-norm of order one ! Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  10. Error Measure 0 0 ( ) ( ) ( ) h ( ) ( ) i d D D J J + u v u v v u u - v u - v = = J J J ; ; ; ; 0 ( ) ( ) ( ) h ( ) i D J J J u v u - v - v u - v = J ; ; • We need a better error measure, stronger than L2, weaker than BV • Possible choice: Bregman distance Bregman67 • Real distance for a strictly convex differentiable functional – not symmetric • Symmetric version Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  11. Bregman Distance 1 2 ( ) k k 1 D ¡ u v u v = J 2 ; ( ) k k J 2 u u = 2 • Bregman distances reduce to known measures for standard energies • Example 1: Subgradient = Gradient = u Bregman distance becomes Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  12. Bregman Distance Z Z u Z Z ( ) ( ) l D + ¡ u v u o g v u = J ; ( ) l J u u o g u - u v = • Example 2: Subgradient = Gradient = logu Bregman distance becomes Kullback-Leibler divergence (relative Entropy) Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  13. Bregman Distance ( ) @ J 2 p v p ( ( ) ) ( h ) ( ) h i i d D J J u v u v u p - - p v - u - p v u - v = = J J u v ; ; ; ; • Total variation is neither symmetric nor differentiable • Define generalized Bregman distance for each subgradient • Symmetric version Kiwiel 97, Chen-Teboulle 97 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  14. Bregman Distance p ( ( ) ) h ( ) i h i ( ) ( ) @ @ D J J J J 2 2 u v v p u v - p p v p v v = = J ; ; ; ; ; • For energies homogeneous of degree one, we have • Bregman distance becomes Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  15. Bregman Distance ( ( ) ) d f 6 0 u u u v = = T V ; • Bregman distance for total variation is not a strict distance, can be zero for • In particular dTV is zero for contrast change Resmerita-Scherzer 06 • Bregman distance is still not negative (convexity) • Bregman distance can provide information about edges Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  16. Error Estimation 2 j j ( ) ( ) j j ¸ @ ¸ f ¸ f f @ L ­ + + \ 2 2 p p q u - q u - p - u q = = T V T V ; • For estimate in terms of l we need smoothness condition on data • Optimality condition for ROF Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  17. Error Estimation 2 k k q 1 ¡ h ( ) i h i ¸ f f f ( ) h i ( ) d f f ¸ + ¡ O · u - p - q u - q u = u p - q u - = = T V ; ; ; ; ¸ 4 2 k k q 2 k k ¸ f · + ¡ u ¸ 4 • Apply to u – v • Estimate for Bregman distance, mb-Osher 04 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  18. Error Estimation 2 k k ¸ q 2 ( ) h i k k d f f f · + u p - q u - - g = T V 2 ; ; j j ( ) ¸ @ 4 2 L ­ \ 2 q g T V 1 ¡ k k ¸ f g - » • In practice we have to deal with noisy data f(perturbation of some exact data g) • Analogous estimate for Bregman distance • Optimal choice of the parameter i.e. of the order of the noise variance Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  19. Error Estimation 2 j j ( ) @ L ­ \ 2 q g T V • Analogous estimate for TV flow mb-Resmerita-He 07 • Regularization parameter is stopping time T of the flowT ~ l-1 • Note: all estimates multivalued ! Hold for any subgradient satisfying Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  20. Interpretation ( ( ) ( ( ) ( ) ) ) ² ² ² ² ² Ã Ã Ã Ã d d r r 0 1 0 1 · · ¡ ½ ¢ q s u p p ² ² = = i i ; ; • Let g be piecewise constant with white background and color values ci on regions Wi • Then we obtain subgradients of the form with signed distance function di and y s.t. Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  21. Interpretation ( ( ) ) ² ² à d d r r ¢ q = i i 2 0 ( ) ( ) ( ) j j ² ² à d à d d ¢ r ¢ 0 0 + = = i i i • e chosen smaller than distance between two region boundaries • Note: on the region boundary (di = 0) subgradient equals mean curvature of edge Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  22. Interpretation Z ² q ( ) j j ² l f d @ D r ­ i i ¸ q ( ) ( ( ) ) ² Ã d d m n q u g o n u D r r = ( n S ) i i @ T V ­ ­ T V ¢ u g s u p u q - = ; i i i T V ; ² k k · 1 q 1 • Bregman distances given by • If we only take the sup over those g with and let e tend to zero we obtain Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  23. Interpretation • Multivalued error estimates imply quantitative estimate for total variation of u away from the discontinuity set of g • Other geometric estimates possible by different choice of subgradients, different limits Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  24. Extensions ¸ j j ¤ @ A 2 2 k k j j q w g f A i = T V ¡ + u u m n ! T V 2 B V 2 u • Direct extension to deconvolution / linear inverse problems: A linear operator under standard source condition mb-Osher 04 • Nonlinear problemsResmerita-Scherzer 06, Hofmann-Kaltenbacher-Pöschl-Scherzer 07 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  25. Extensions • Stronger estimates under stronger conditionsResmerita 05 • Numerical analysis for appropriate discretizations (correct discretization of subgradient crucial) mb 07 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  26. Future Tasks • Extension to other fitting functionals (relative entropy, log-likelihood functionals for different noise models) • Extension to anisotropic TV (Interpretation of subgradients) • Extension to geometric problems (segmentation by Chan-Vese, Mumford-Shah): use exact relaxation in BV with bound constraintsChan-Esedoglu-Nikolova 04 Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

  27. Download and Contact Papers and talks at www.math.uni-muenster.de/u/burger Email martin.burger@uni-muenster.de Error Estimation in TV Imaging ICIAM 07, Zürich, July 07

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