# Mumford-Shah Loss Functional for Image Segmentation with Deep Learning

## TL;DR

This paper introduces a Mumford-Shah based loss function for deep learning image segmentation, enabling semi-supervised, unsupervised, and regularized training, reducing dependence on large labeled datasets.

## Contribution

The paper proposes a novel Mumford-Shah inspired loss function for deep neural networks, bridging classical energy-based segmentation with modern deep learning methods.

## Key findings

- Effective in semi-supervised and unsupervised segmentation
- Enhances supervised segmentation with regularization
- Demonstrates improved results on multiple datasets

## Abstract

Recent state-of-the-art image segmentation algorithms are mostly based on deep neural networks, thanks to their high performance and fast computation time. However, these methods are usually trained in a supervised manner, which requires large number of high quality ground-truth segmentation masks. On the other hand, classical image segmentation approaches such as level-set methods are formulated in a self-supervised manner by minimizing energy functions such as Mumford-Shah functional, so they are still useful to help generation of segmentation masks without labels. Unfortunately, these algorithms are usually computationally expensive and often have limitation in semantic segmentation. In this paper, we propose a novel loss function based on Mumford-Shah functional that can be used in deep-learning based image segmentation without or with small labeled data. This loss function is based on the observation that the softmax layer of deep neural networks has striking similarity to the characteristic function in the Mumford-Shah functional. We show that the new loss function enables semi-supervised and unsupervised segmentation. In addition, our loss function can be also used as a regularized function to enhance supervised semantic segmentation algorithms. Experimental results on multiple datasets demonstrate the effectiveness of the proposed method.