# Selection of the Regularization Parameter in the Ambrosio-Tortorelli Approximation of the Mumford-Shah Functional for Image Segmentation

## TL;DR

This paper analyzes how the regularization parameter affects the segmentation performance of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional, proposing a strategy for optimal parameter selection and demonstrating improved results on real images.

## Contribution

It provides an asymptotic analysis of the Ambrosio-Tortorelli functional's behavior and introduces a new parameter selection method for better image segmentation.

## Key findings

- Segmentation ability varies with the regularization parameter.
- The functional loses segmentation ability as the parameter approaches zero.
- The proposed parameter selection improves segmentation results on real images.

## Abstract

The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah functional that has been widely used for image segmentation. The approximation has the advantages of being easy to implement, maintaining the segmentation ability, and $\Gamma$-converging to the Mumford-Shah functional. However, it has been observed in actual computation that the segmentation ability of the Ambrosio-Tortorelli functional varies significantly with different values of the parameter and it even fails to $\Gamma$-converge to the original functional for some cases. In this paper we present an asymptotic analysis on the gradient flow equation of the Ambrosio-Tortorelli functional and show that the functional can have different segmentation behavior for small but finite values of the regularization parameter and eventually loses its segmentation ability as the parameter goes to zero when the input image is treated as a continuous function. This is consistent with the existing observation as well as the numerical examples presented in this work. A selection strategy for the regularization parameter and a scaling procedure for the solution are devised based on the analysis. Numerical results show that they lead to good segmentation of the Ambrosio-Tortorelli functional for real images.