Total variation minimization with L1 data fidelity as a contrast invariant filter

This paper sheds new light on minimization of the total variation under the L¹-norm as data fidelity term (L¹ +TV) and its link with mathematical morphology. It is well known that morphological filters feature the property of being invariant with respect to any change of contrast. First, we show that minimization of L¹ + 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 that demonstrate that these new filters can be particularly useful as a pre-processing stage before segmentation.