Note on the Existence of Minimizers for Variational Geometric Active Contours

TL;DR

This paper proves the existence of minimizers for a shape-based segmentation functional using level set methods, ensuring convergence of numerical algorithms and applicability to various segmentation models.

Contribution

It provides a rigorous proof of minimizer existence for a shape-informed segmentation functional with level sets, enhancing reliability of segmentation algorithms.

Findings

Existence of minimizers guarantees convergence of numerical methods.

The proof applies broadly to other segmentation models.

Supports the development of reliable shape-based segmentation algorithms.

Abstract

We propose here a proof of existence of a minimizer of a segmentation functional based on a priori information on target shapes, and formulated with level sets. The existence of a minimizer is very important, because it guarantees the convergence of any numerical methods (either gradient descents techniques and variants, or PDE resolutions) used to solve the segmentation model. This work can also be used in many other segmentation models to prove the existence of a minimizer.