# A Note on the Discrete Binary Mumford-Shah Model

## Abstract

This paper is concerned itself with the analysis of the two-phase Mumford-Shah model also known as the active contour without edges model introduced by Chan and Vese. It consists of approximating an observed image by a piecewise constant image which can take only two values. First we show that this model with the ${\displaystyle L^{1}}$-norm as data fidelity yields a contrast invariant filter which is a well known property of morphological filters. Then we consider a discrete version of the original problem. We show that an inclusion property holds for the minimizers. The latter is used to design an efficient graph-cut based algorithm which computes an exact minimizer. Some preliminary results are presented.

## Bibtex (lrde.bib)

```@InProceedings{	  darbon.07.mirage,
author	= {J\'er\^ome Darbon},
title		= {A Note on the Discrete Binary {Mumford-Shah} Model},
booktitle	= {Proceedings of the international Computer Vision /
Computer Graphics Collaboration Techniques and Applications
(MIRAGE 2007)},
year		= 2007,
month		= mar,
abstract	= {This paper is concerned itself with the analysis of the
two-phase Mumford-Shah model also known as the active
contour without edges model introduced by Chan and Vese. It
consists of approximating an observed image by a piecewise
constant image which can take only two values. First we
show that this model with the \$L^1\$-norm as data fidelity
yields a contrast invariant filter which is a well known
property of morphological filters. Then we consider a
discrete version of the original problem. We show that an
inclusion property holds for the minimizers. The latter is
used to design an efficient graph-cut based algorithm which
computes an exact minimizer. Some preliminary results are
presented.}
}```