# A Vectorial Self-Dual Morphological Filter based on Total Variation Minimization

## Abstract

We present a vectorial self dual morphological filter. Contrary to many methods, our approach does not require the use of an ordering on vectors. It relies on the minimization of the total variation with ${\displaystyle L^{1}}$ norm as data fidelity on each channel. We further constraint this minimization in order not to create new values. It is shown that this minimization yields a self-dual and contrast invariant filter. Although the above minimization is not a convex problem, we propose an algorithm which computes a global minimizer. This algorithm relies on minimum cost cut-based optimizations.

## Bibtex (lrde.bib)

```@InProceedings{	  darbon.05.isvc,
author	= {J\'er\^ome Darbon and Sylvain Peyronnet},
title		= {A Vectorial Self-Dual Morphological Filter based on Total
Variation Minimization},
booktitle	= {Proceedings of the First International Conference on
Visual Computing},
year		= 2005,
month		= dec,
pages		= {388--395},
series	= {Lecture Notes in Computer Science Series},
publisher	= {Springer-Verlag},
volume	= 3804,
abstract	= {We present a vectorial self dual morphological filter.
Contrary to many methods, our approach does not require the
use of an ordering on vectors. It relies on the
minimization of the total variation with \$L^1\$ norm as data
fidelity on each channel. We further constraint this
minimization in order not to create new values. It is shown
that this minimization yields a self-dual and contrast
invariant filter. Although the above minimization is not a
convex problem, we propose an algorithm which computes a
global minimizer. This algorithm relies on minimum cost
cut-based optimizations.}
}```