# Approximation of the Mumford-Shah Functional by Phase Fields of Bounded Variation

## TL;DR

This paper introduces a novel phase field approximation of the Mumford-Shah functional using bounded variation functions, enabling sharper image segmentation results compared to traditional methods.

## Contribution

The paper proposes a new phase field model based on BV functions for Mumford-Shah approximation, enhancing image segmentation quality.

## Key findings

- The BV-based approximation produces sharper phase fields.
- Numerical methods incorporate total variation minimization.
- Comparison shows improved segmentation sharpness.

## Abstract

In this paper we introduce a new phase field approximation of the Mumford-Shah functional similar to the well-known one from Ambrosio and Tortorelli. However, in our setting the phase field is allowed to be a function of bounded variation, instead of an $H^1$-function. In the context of image segmentation, we also show how this new approximation can be used for numerical computations, which contains a total variation minimization of the phase field variable, as it appears in many problems of image processing. A comparison to the classical Ambrosio-Tortorelli approximation, where the phase field is an $H^1$-function, shows that the new model leads to sharper phase fields.