# Total Variation Minimization with L^1 Data Fidelity as a Contrast Invariant Filter

## Abstract

This paper sheds new light on minimization of the total variation under the ${\displaystyle L^{1}}$-norm as data fidelity term (${\displaystyle L^{1}+TV}$) and its link with mathematical morphology. It is well known that morphological filters enjoy the property of being invariant with respect to any change of contrast. First, we show that minimization of ${\displaystyle L^{1}+TV}$ yields a self-dual and contrast invariant filter. Then, we further constrain the minimization process by only optimizing the grey levels of level sets of the image while keeping their boundaries fixed. This new constraint is maintained thanks to the Fast Level Set Transform which yields a complete representation of the image as a tree. We show that this filter can be expressed as a Markov Random Field on this tree. Finally, we present some results which demonstrate that these new filters can be particularly useful as a preprocessing stage before segmentation.

## Bibtex (lrde.bib)

```@InProceedings{	  darbon.05.ispa,
author	= {J\'er\^ome Darbon},
title		= {Total Variation Minimization with \$L^1\$ Data Fidelity as a
Contrast Invariant Filter},
booktitle	= {Proceedings of the 4th International Symposium on Image
and Signal Processing and Analysis (ISPA 2005)},
year		= 2005,
month		= sep,
pages		= {221--226},
abstract	= {This paper sheds new light on minimization of the total
variation under the \$L^1\$-norm as data fidelity term
(\$L^1+TV\$) and its link with mathematical morphology. It is
well known that morphological filters enjoy the property of
being invariant with respect to any change of contrast.
First, we show that minimization of \$L^1+TV\$ yields a
self-dual and contrast invariant filter. Then, we further
constrain the minimization process by only optimizing the
grey levels of level sets of the image while keeping their
boundaries fixed. This new constraint is maintained thanks
to the Fast Level Set Transform which yields a complete
representation of the image as a tree. We show that this
filter can be expressed as a Markov Random Field on this
tree. Finally, we present some results which demonstrate
that these new filters can be particularly useful as a
preprocessing stage before segmentation.}
}```