1 Total variation minimization Numerical Analysis, Error Estimation, and Extensions Martin Burger...
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Total variation minimization Numerical Analysis, Error Estimation, and Extensions
Martin Burger
Johannes Kepler University LinzSFB Numerical-Symbolic-Geometric Scientific ComputingRadon Institute for Computational & Applied Mathematics
Westfälische Wilhelms Universität Münster
Total variation minimization
Obergurgl, September 2006 2
Stan Osher, Jinjun Xu, Guy Gilboa (UCLA)
Lin He (Linz / UCLA)
Klaus Frick, Otmar Scherzer (Innsbruck)
Carola Schönlieb (Vienna)
Don Goldfarb, Wotao Yin (Columbia)
Collaborations
Total variation minimization
Obergurgl, September 2006 3
Total variation methods are popular in imaging (and inverse problems), since
- they keep sharp edges- eliminate oscillations (noise)- create new nice mathematics
Many related approaches appeared in the last years, e.g. ℓ 1 penalization / sparsity techniques
Introduction
Total variation minimization
Obergurgl, September 2006 4
Total variation and related methods have some shortcomings
- difficult to analyze and to obtain error estimates- systematic errors (clean images not reconstructed perfectly)- computational challenges- some extensions to other imaging tasks are not well understood (e.g. inpainting)
Introduction
Total variation minimization
Obergurgl, September 2006 5
Starting point of the analysis is the ROF model for denoising
Rudin-Osher Fatemi 89/92, Acar-Vogel 93, Chambolle-Lions 96, Vogel 95/96, Scherzer-Dobson 96, Chavent-Kunisch 98, Meyer 01,…
ROF Model
Total variation minimization
Obergurgl, September 2006 6
ROF ModelReconstruction (code by Jinjun Xu)
clean noisy ROF
Total variation minimization
Obergurgl, September 2006 7
First question for error estimation: estimate difference of u (minimizer of ROF) and f in terms of
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 !
Error Estimation
Total variation minimization
Obergurgl, September 2006 8
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
Error Estimation
Total variation minimization
Obergurgl, September 2006 9
Total variation is neither symmetric nor differentiable Define generalized Bregman distance for each subgradient
Symmetric version
Kiwiel 97, Chen-Teboulle 97
Error Estimation
Total variation minimization
Obergurgl, September 2006 10
Since TV seminorm is homogeneous of degree one, we have
Bregman distance becomes
Error Estimation
Total variation minimization
Obergurgl, September 2006 11
Bregman distance for TV 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 (TV convex) Bregman distance can provide information about edges
Error Estimation
Total variation minimization
Obergurgl, September 2006 12
Let v be piecewise constant with white background and color values on regions Then we obtain subgradients of the form
with signed distance function and
Error Estimation
Total variation minimization
Obergurgl, September 2006 13
Bregman distances given by
In the limit we obtain for being piecewise continuous
Error Estimation
Total variation minimization
Obergurgl, September 2006 14
For estimate in terms of we need smoothness condition on data
Optimality condition for ROF
Error Estimation
Total variation minimization
Obergurgl, September 2006 15
Subtract q
Estimate for Bregman distance, mb-Osher 04
Error Estimation
Total variation minimization
Obergurgl, September 2006 16
In practice we have to deal with noisy data f (perturbation of some exact data g)
Estimate for Bregman distance
Error Estimation
Total variation minimization
Obergurgl, September 2006 17
Optimal choice of the penalization parameter
i.e. of the order of the noise variance
Error Estimation
Total variation minimization
Obergurgl, September 2006 18
Direct extension to deconvolution / linear inverse problems
under standard source condition
mb-Osher 04 Extension: stronger estimates under stronger conditions, Resmerita 05
Nonlinear inverse problems, Resmerita-Scherzer 06
Error Estimation
Total variation minimization
Obergurgl, September 2006 19
Natural choice: primal discretization with piecewise constant functions on grid
Problem 1: Numerical analysis (characterization of discrete subgradients) Problem 2: Discrete problems are the same for any anisotropic version of the total variation
Discretization
Total variation minimization
Obergurgl, September 2006 20
In multiple dimensions, nonconvergence of the primal discretization for the isotropic TV (p=2) can be shown
Convergence of anisotropic TV (p=1) on rectangular aligned grids Fitzpatrick-Keeling 1997
Discretization
Total variation minimization
Obergurgl, September 2006 21
Alternative: perform primal-dual discretization for optimality system (variational inequality)
with convex set
Primal-Dual Discretization
Total variation minimization
Obergurgl, September 2006 22
Discretization
Discretized convex set with appropriate elements (piecewise linear in 1D, Raviart-Thomas in multi-D)
Primal-Dual Discretization
Total variation minimization
Obergurgl, September 2006 23
In 1 D primal, primal-dual, and dual discretization are equivalent Error estimate for Bregman distance by analogous techniques
Note that only the natural condition is needed to show
Primal / Primal-Dual Discretization
Total variation minimization
Obergurgl, September 2006 24
In multi-D similar estimates, additional work since projection of subgradient is not discrete subgradient.
Primal-dual discretization equivalent to discretized dual minimization (Chambolle 03,
Kunisch-Hintermüller 04). Can be used for existence of discrete solution, stability of p
mb 06/07 ?
Primal / Primal-Dual Discretization
Total variation minimization
Obergurgl, September 2006 25
For most imaging applications Cartesian grids are used. Primal dual discretization can be reinterpreted as a finite difference scheme in this setup. Value of image intensity corresponds to color in a pixel of width h around the grid point. Raviart-Thomas elements on Cartesian grids particularly easy. First component piecewise linear in x, pw constant in y,z, etc. Leads to simple finite difference scheme with staggered grid
Cartesian Grids
Total variation minimization
Obergurgl, September 2006 26
ROF minimization has a systematic error, total variation of the reconstruction is smaller than total variation of clean image. Image features left in residual f-u
g, clean f, noisy u, ROF f-u
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 27
Idea: add the residual („noise“) back to the image to pronounce the features decreased to much. Then do ROF again. Iterative procedure
Osher-mb-Goldfarb-Xu-Yin 04
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 28
Improves reconstructions significantly
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 30
Simple observation from optimality condition
Consequently, iterative refinement equivalent to Bregman iteration
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 31
Choice of parameter less important, can be kept small (oversmoothing). Regularizing effect comes from appropriate stopping. Quantitative stopping rules available, or „stop when you are happy“ – S.O. Limit to zero can be studied. Yields gradient flow for the dual variable („inverse scale space“)
mb-Gilboa-Osher-Xu 06, mb-Frick-Osher-Scherzer 06
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 32
Non-quadratic fidelity is possible, some caution needed for L1 fidelityHe-mb-Osher 05, mb-Frick-Osher-Scherzer 06
Error estimation in Bregman distance mb-Resmerita 06, in prep
Further details see talk of Klaus Frick
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 33
Extension I: Inverse Scale Space Movie by M. Bachmayr, Master Thesis 06
step.avi.lnk
Total variation minimization
Obergurgl, September 2006 34
Application to other regularization techniques, e.g. wavelet thresholding is straightforward
Starting from soft shrinkage, iterated refinement yields firm shrinkage, inverse scale space becomes hard shrinkageOsher-Xu 06
Bregman distance natural sparsity measure, source condition just requires sparse signal, number of nonzero components is smoothness measure in error estimates
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 35
Total variation, inverse scale space, and shrinkage techniques can be combined nicely See talk by Lin He
Extension I: Iterative Refinement & ISS
Total variation minimization
Obergurgl, September 2006 36
Total variation will prefer isotropic structures (circles, spheres) or special anisotropies
In many applications one wants sharp corners in different directions. Adaptive anisotropy is needed
Can be incorporated in ROF and ISS. See talk by Benjamin Berkels
Extension II: Anisotropy
Total variation minimization
Obergurgl, September 2006 37
Difficult to construct total variation techniques for inpainting Original extensions of ROF failed to obtain natural connectivity (see book by Chan, Shen 05)
Inpainting region , image f (noisy) given on Try to minimize
Extension III: Inpainting
Total variation minimization
Obergurgl, September 2006 38
Optimality condition will have the form
with A being a linear operator defining the norm
In particular p = 0 in D !
Extension III: Inpainting
Total variation minimization
Obergurgl, September 2006 39
Different iterated approach (motivated by Cahn-Hilliard inpainting, Bertozzi et al 05) Minimize in each step
First term for damping, second for fidelity (fit to f where given, and to old iterate in the inpainting region), third term for smoothing
Extension III: Inpainting
Total variation minimization
Obergurgl, September 2006 40
Continuous flow for damping parameter to zero
Fourth order flow for H-1 norm
Stationary solution (existence ?) satisfies
Extension III: Inpainting
Total variation minimization
Obergurgl, September 2006 41
Result: Penguins
Extension III: Inpainting
Total variation minimization
Obergurgl, September 2006 42
Original motivation: Osher-Marquinha 01 used preconditioned gradient flow for ROF
Stationary state assumed to be ROF minimizer
Computational observation: not always true !
Trivial observation: for initial value u(0) = 0 the flow remains zero for all time !
Extension IV: Manifolds
Total variation minimization
Obergurgl, September 2006 43
Embarrassing observation: flow always created by transport from initial value
Important observation: Stationary state minimizes ROF on the manifold
Extension IV: Manifolds
Total variation minimization
Obergurgl, September 2006 44
Surprising observation: for f being the indicator function of a convex set, the flow is equivalent to the gradient flow of the L1 version of ROF
No loss of contrast ! More detailed analysis for general images needed Possible extension to ROF minimization on other manifolds by metric gradient flows
Extension IV: Manifolds
Total variation minimization
Obergurgl, September 2006 45
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Papers and Talks:www.indmath.uni-linz.ac.at/people/burger
from October: wwwmath1.uni-muenster.de/num
e-mail: [email protected]