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

arXiv: https://arxiv.org/abs/1904.02872 · HTML: https://mlsanity.com/paper/1904.02872/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2019.

## Built on by (most cited first, 30 of 35)

- [Scribble-Supervised Medical Image Segmentation via Dual-Branch Network and Dynamically Mixed Pseudo Labels Supervision](https://mlsanity.com/paper/2203.02106/) (2022)
- [Box-supervised Instance Segmentation with Level Set Evolution](https://mlsanity.com/paper/2207.09055/) (2022)
- [Overview of the HECKTOR Challenge at MICCAI 2021: Automatic Head and Neck Tumor Segmentation and Outcome Prediction in PET/CT Images](https://mlsanity.com/paper/2201.04138/) (2022)
- [Weak-Mamba-UNet: Visual Mamba Makes CNN and ViT Work Better for Scribble-based Medical Image Segmentation](https://mlsanity.com/paper/2402.10887/) (2024)
- [ScribFormer: Transformer Makes CNN Work Better for Scribble-based Medical Image Segmentation](https://mlsanity.com/paper/2402.02029/) (2024)
- [ScribbleVC: Scribble-supervised Medical Image Segmentation with Vision-Class Embedding](https://mlsanity.com/paper/2307.16226/) (2023)
- [Box2Mask: Box-supervised Instance Segmentation via Level-set Evolution](https://mlsanity.com/paper/2212.01579/) (2022)
- [PA-Seg: Learning from Point Annotations for 3D Medical Image Segmentation using Contextual Regularization and Cross Knowledge Distillation](https://mlsanity.com/paper/2208.05669/) (2022)
- [PyMIC: A deep learning toolkit for annotation-efficient medical image segmentation](https://mlsanity.com/paper/2208.09350/) (2022)
- [Weakly supervised segmentation with point annotations for histopathology images via contrast-based variational model](https://mlsanity.com/paper/2304.03572/) (2023)
- [Ultrasound Nodule Segmentation Using Asymmetric Learning with Simple Clinical Annotation](https://mlsanity.com/paper/2404.14852/) (2024)
- [Scribble-based 3D Multiple Abdominal Organ Segmentation via Triple-branch Multi-dilated Network with Pixel- and Class-wise Consistency](https://mlsanity.com/paper/2309.09730/) (2023)
- [SAM-Driven Weakly Supervised Nodule Segmentation with Uncertainty-Aware Cross Teaching](https://mlsanity.com/paper/2407.13553/) (2024)
- [Deep Superpixel Cut for Unsupervised Image Segmentation](https://mlsanity.com/paper/2103.06031/) (2021)
- [Annotation by Clicks: A Point-Supervised Contrastive Variance Method for Medical Semantic Segmentation](https://mlsanity.com/paper/2212.08774/) (2022)
- [Learning Fuzzy Clustering for SPECT/CT Segmentation via Convolutional Neural Networks](https://mlsanity.com/paper/2104.08623/) (2021)
- [PCLMix: Weakly Supervised Medical Image Segmentation via Pixel-Level Contrastive Learning and Dynamic Mix Augmentation](https://mlsanity.com/paper/2405.06288/) (2024)
- [Cell Instance Segmentation: The Devil Is in the Boundaries](https://mlsanity.com/paper/2510.09848/) (2025)
- [Segmentation and Risk Score Prediction of Head and Neck Cancers in PET/CT Volumes with 3D U-Net and Cox Proportional Hazard Neural Networks](https://mlsanity.com/paper/2202.07823/) (2022)
- [Pixel-Level Clustering Network for Unsupervised Image Segmentation](https://mlsanity.com/paper/2310.16234/) (2023)
- [EAUWSeg: Eliminating annotation uncertainty in weakly-supervised medical image segmentation](https://mlsanity.com/paper/2501.01658/) (2025)
- [Deep Level Set for Box-supervised Instance Segmentation in Aerial Images](https://mlsanity.com/paper/2112.03451/) (2021)
- [Estimating Appearance Models for Image Segmentation via Tensor Factorization](https://mlsanity.com/paper/2208.07853/) (2022)
- [Unsupervised Cardiac Segmentation Utilizing Synthesized Images from Anatomical Labels](https://mlsanity.com/paper/2301.06043/) (2023)
- [Variational multichannel multiclass segmentation using unsupervised lifting with CNNs](https://mlsanity.com/paper/2302.02214/) (2023)
- [Semi-Automated Segmentation of Geoscientific Data Using Superpixels](https://mlsanity.com/paper/2303.11404/) (2023)
- [AttenScribble: Attentive Similarity Learning for Scribble-Supervised Medical Image Segmentation](https://mlsanity.com/paper/2312.06614/) (2023)
- [Segmentation of tibiofemoral joint tissues from knee MRI using MtRA-Unet and incorporating shape information: Data from the Osteoarthritis Initiative](https://mlsanity.com/paper/2401.12932/) (2024)
- [DABS-LS: Deep Atlas-Based Segmentation Using Regional Level Set Self-Supervision](https://mlsanity.com/paper/2402.10425/) (2024)
- [A Bayesian Approach to Weakly-supervised Laparoscopic Image Segmentation](https://mlsanity.com/paper/2410.08509/) (2024)

## Abstract (arXiv)

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.