Image restoration with discrete constrained Total Variation---Part~I: Fast and exact optimization
From LRDE
- Authors
- Jérôme Darbon, Marc Sigelle
- Journal
- Journal of Mathematical Imaging and Vision
- Type
- article
- Keywords
- Image
- Date
- 2006-03-24
Abstract
This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems at each level. Exact solutions of these binary problems are found thanks to minimum cost cut techniques in graphs. These binary solutions are proved to be monotone increasing with levels and yield thus an exact solution of the discrete original problem. Furthermore we show that minimization of total variation under $L^1$ data fidelity term yields a self-dual contrast invariant filter. Finally we present some results.
Bibtex (lrde.bib)
@Article{ darbon.06.jmiv, author = {J\'er\^ome Darbon and Marc Sigelle}, title = {Image restoration with discrete constrained {T}otal {Variation}---Part~{I}: Fast and exact optimization}, journal = {Journal of Mathematical Imaging and Vision}, year = 2006, volume = 26, number = 3, month = dec, pages = {261--276}, project = {Image}, abstract = {This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems at each level. Exact solutions of these binary problems are found thanks to minimum cost cut techniques in graphs. These binary solutions are proved to be monotone increasing with levels and yield thus an exact solution of the discrete original problem. Furthermore we show that minimization of total variation under $L^1$ data fidelity term yields a self-dual contrast invariant filter. Finally we present some results.} } @Article{ darbon.06.jmivb, author = {J\'er\^ome Darbon and Marc Sigelle}, title = {Image restoration with discrete constrained {T}otal {Variation}---Part~{II}: Levelable functions, convex priors and non-convex case}, journal = {Journal of Mathematical Imaging and Vision}, year = 2006, volume = 26, number = 3, month = dec, pages = {277--291}, project = {Image}, abstract = {In Part II of this paper we extend the results obtained in Part I for total variation minimization in image restoration towards the following directions: first we investigate the decomposability property of energies on levels, which leads us to introduce the concept of levelable regularization functions (which TV is the paradigm of). We show that convex levelable posterior energies can be minimized exactly using the level-independant cut optimization scheme seen in part I. Next we extend this graph cut scheme optimization scheme to the case of non-convex levelable energies. We present convincing restoration results for images corrupted with impulsive noise. We also provide a minimum-cost based algorithm which computes a global minimizer for Markov Random Field with convex priors. Last we show that non-levelable models with convex local conditional posterior energies such as the class of generalized gaussian models can be exactly minimized with a generalized coupled Simulated Annealing.} }