Optimal partitions, regularized solutions, and application to image classification

Abstract

This article is concerned with the problem of finding a regular and homogeneous partition of a Lipschitz open set Ω of

. This type of problem occurs in image classification which consists in assigning a label (that is a class or a phase) to each pixel of an observed image. Such a problem can numerically be solved by using a level set formulation. But this approach generally relies on heuristic arguments. We intend here to give a theoretical justification in the two phases case. In the first part of the paper, we prove that the partition problem we consider admits a solution. And in the second part, we justify the numerical approach which is generally used to compute a solution.

Keywords:

• Sets with finite perimeter
• BV
• Perturbation
• Partial differential equations
• Image processing
• AMS Subject Classifications: 35J
• 49J
• 68U

Notes

E-mail: [email protected]

Additional information

Notes on contributors

Jean-Franç Aujol ‡

E-mail: [email protected]