# Exact optimization of discrete constrained total variation minimization problems

### From LRDE

- Authors
- Jérôme Darbon, Marc Sigelle
- Place
- Paris, France
- Type
- techreport
- Projects
- Olena
- Keywords
- Image
- Date
- 2004-10-01

## Abstract

This paper deals with the total variation minimization problem when the fidelity is either the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2}**
-norm or the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^1}**
-norm. We propose an algorithm which computes the exact solution of these two problems after discretization. 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 associated with each level set. Exact solutions of these binary problems are found thanks to minimum-cut techniques. We prove that these binary solutions are increasing and thus allow to reconstruct the solution of the original problems.

## Bibtex (lrde.bib)

@TechReport{ darbon.04.tr, author = {J\'er\^ome Darbon and Marc Sigelle}, title = {Exact optimization of discrete constrained total variation minimization problems}, institution = {ENST}, year = 2004, number = {2004C004}, address = {Paris, France}, month = oct, annote = {This technical report corresponds to the publication darbon.04.iwcia. ; 200412-IWCIA}, abstract = {This paper deals with the total variation minimization problem when the fidelity is either the $L^2$-norm or the $L^1$-norm. We propose an algorithm which computes the exact solution of these two problems after discretization. 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 associated with each level set. Exact solutions of these binary problems are found thanks to minimum-cut techniques. We prove that these binary solutions are increasing and thus allow to reconstruct the solution of the original problems.} }