Error Estimation in TV Imaging

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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

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Joint Work with

• Stan Osher (UCLA)
• mb-Osher, Inverse Problems 04
• Elena Resmerita, Lin He (Linz)
• mb-Resmerita-He, Computing 07

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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

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TV Imaging

• Total variation is a convex, but not
  differentiable and not strictly convex
  functional
• 
  
• Banach space BV consisting of all L1 functions
  of bounded variation

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Denoising Models

• 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, ..

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ROF Model

• 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

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ROF Model
Reconstruction (code by Jinjun
Xu) clean noisy
ROF
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ROF Model

• ROF model denoises cartoon images resp. computes
  the cartoon of an arbitrary image, natural
  spatial multi-scale decomposition by varying l

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Error Estimation ?

• First question for error estimation estimate
  difference of u (minimizer of ROF) and f in terms
  of l
• Estimate in the L2 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 !

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Error Measure

• We need a better error measure, stronger than
  L2, weaker than BV
• Possible choice Bregman distance Bregman 67
• Real distance for a strictly convex
  differentiable functional not symmetric
• Symmetric version

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Bregman Distance

• Bregman distances reduce to known measures for
  standard energies
• Example 1
• Subgradient Gradient u
• Bregman distance becomes

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Bregman Distance

• Example 2
• Subgradient Gradient log u
• Bregman distance becomes Kullback-Leibler
  divergence (relative Entropy)

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Bregman Distance

• Total variation is neither symmetric nor
  differentiable
• Define generalized Bregman distance for each
  subgradient
• Symmetric version
• Kiwiel 97, Chen-Teboulle 97

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Bregman Distance

• For energies homogeneous of degree one, we have
• Bregman distance becomes

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Bregman Distance

• 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

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Error Estimation

• For estimate in terms of l we need smoothness
  condition on data
• Optimality condition for ROF

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Error Estimation

• Apply to u v
• Estimate for Bregman distance, mb-Osher 04

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Error Estimation

• 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

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Error Estimation

• Analogous estimate for TV flow mb-Resmerita-He
  07
• Regularization parameter is stopping time T of
  the flow T l-1
• Note all estimates multivalued ! Hold for any
  subgradient satisfying

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Interpretation

• 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.

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Interpretation

• e chosen smaller than distance between two
  region boundaries
• Note on the region boundary (di 0)
• subgradient equals mean curvature of edge

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Interpretation

• Bregman distances given by
• If we only take the sup over those g with and
  let e tend to zero we obtain

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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

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Extensions

• 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

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Extensions

• Stronger estimates under stronger conditions
  Resmerita 05
• Numerical analysis for appropriate
  discretizations (correct discretization of
  subgradient crucial) mb 07

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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 constraints Chan-Esedoglu-Nikolova
  04

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Download and Contact

• Papers and talks at
• www.math.uni-muenster.de/u/burger
• Email
• martin.burger_at_uni-muenster.de