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

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

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 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 TV Imaging

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

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 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,.. Denoising Models u ( t = 0 ) = f ¸ 2 Z ( u ¡ f ) 2 + j u j TV ! m i n u 2 t u = r ¢ ( r u j r u j ) 2 j u j TV

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 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 ROF J ( u ) = f p 2 X ¤ j 8 v 2 X : J ( u ) + h p ; v ¡ u i · J ( v )g p + ¸ u = ¸f ; p j u j TV

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 ROF Model Reconstruction (code by Jinjun Xu) cleannoisy ROF

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

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 First question for error estimation: estimate difference of u (minimizer of ROF) and f in terms of Estimate in the L 2 norm 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 ?

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 We need a better error measure, stronger than L 2, weaker than BV Possible choice: Bregman distance Bregman 67 Real distance for a strictly convex differentiable functional – not symmetric Symmetric version Error Measure D J ( u ; v ) = J ( u ) - J ( v ) - h J 0 ( v ) ; u-v i d J ( u ; v ) = D J ( u ; v ) + D J ( v ; u ) = h J 0 ( u ) - J 0 ( v ) ; u-v i

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

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 Example 2: Subgradient = Gradient = log u Bregman distance becomes Kullback-Leibler divergence (relative Entropy) Bregman Distance D J ( u ; v ) = Z u l og u v + Z ( v ¡ u ) J ( u ) = Z u l ogu- Z u

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

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

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

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 For estimate in terms of we need smoothness condition on data Optimality condition for ROF Error Estimation q j f j TV \ L 2 ( ­ ) p + ¸ u = ¸f ; p j u j TV p-q + ¸ ( u- f ) = -q

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

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 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 d TV ( u ; f ) = h p-q ; u- f i · k q k 2 4 ¸ + ¸ 2 k f -g k 2 ¸ ¡ 1 » k g- f k q j g j TV \ L 2 ( ­ )

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

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 Let g be piecewise constant with white background and color values c i on regions  i Then we obtain subgradients of the form with signed distance function d i and  s.t. Interpretation q ² = r ¢ ( Ã ² ( d i ) r d i ) 0 · Ã ² · 1 ; Ã ² ( 0 ) = 1 supp ( Ã ² ) ½ ( ¡ ² ; ² )

Error Estimation in TV ImagingICIAM 07, Zürich, July 07  chosen smaller than distance between two region boundaries Note: on the region boundary (d i = 0) subgradient equals mean curvature of edge Interpretation q ² = r ¢ ( Ã ² ( d i ) r d i ) = Ã ² ( 0 ) ¢ d i + ( Ã ² ) 0 ( 0 )j r d i j 2 = ¢ d i

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 Bregman distances given by If we only take the sup over those g with and let  tend to zero we obtain Interpretation D q ² TV ( u ; g ) = sup k q k 1 · 1 Z u r ¢ ( q- Ã ² ( d i ) r d i ) q = r d i ­ i l i m i n f ² D q ² TV ( u ; g ) ¸ j u j TV ( ­ n ­ i )

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 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 Interpretation

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 Direct extension to deconvolution / linear inverse problems: A linear operator under standard source condition mb-Osher 04 Nonlinear problems Resmerita-Scherzer 06, Hofmann-Kaltenbacher-Pöschl-Scherzer 07 Extensions ¸ 2 k A u ¡ f k 2 + j u j TV ! m i n u 2 BV q = A ¤ w j g j TV

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 Stronger estimates under stronger conditions Resmerita 05 Numerical analysis for appropriate discretizations (correct discretization of subgradient crucial) mb 07 Extensions

Error Estimation in TV ImagingICIAM 07, Zürich, July 07 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 constraints Chan-Esedoglu-Nikolova 04 Future Tasks

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