# Exact optimization of discrete constrained total variation minimization problems

## Abstract

This paper deals with the total variation minimization problem when the fidelity is either the $\displaystyle L^2$ -norm or the $\displaystyle L^1$ -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.}
}```