# Learning Euler's Elastica Model for Medical Image Segmentation

arXiv: https://arxiv.org/abs/2011.00526 · HTML: https://mlsanity.com/paper/2011.00526/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2020.

## Built on by (most cited first, 6 of 6)

- [Momentum Contrastive Voxel-wise Representation Learning for Semi-supervised Volumetric Medical Image Segmentation](https://mlsanity.com/paper/2105.07059/) (2021)
- [nnSAM: Plug-and-play Segment Anything Model Improves nnUNet Performance](https://mlsanity.com/paper/2309.16967/) (2023)
- [SHREC 2022: pothole and crack detection in the road pavement using images and RGB-D data](https://mlsanity.com/paper/2205.13326/) (2022)
- [Adaptation of Distinct Semantics for Uncertain Areas in Polyp Segmentation](https://mlsanity.com/paper/2405.07523/) (2024)
- [LGPS: A Lightweight GAN-Based Approach for Polyp Segmentation in Colonoscopy Images](https://mlsanity.com/paper/2503.18294/) (2025)
- [CNN-based Euler's Elastica Inpainting with Deep Energy and Deep Image Prior](https://mlsanity.com/paper/2207.07921/) (2022)

## Abstract (arXiv)

Image segmentation is a fundamental topic in image processing and has been studied for many decades. Deep learning-based supervised segmentation models have achieved state-of-the-art performance but most of them are limited by using pixel-wise loss functions for training without geometrical constraints. Inspired by Euler's Elastica model and recent active contour models introduced into the field of deep learning, we propose a novel active contour with elastica (ACE) loss function incorporating Elastica (curvature and length) and region information as geometrically-natural constraints for the image segmentation tasks. We introduce the mean curvature i.e. the average of all principal curvatures, as a more effective image prior to representing curvature in our ACE loss function. Furthermore, based on the definition of the mean curvature, we propose a fast solution to approximate the ACE loss in three-dimensional (3D) by using Laplace operators for 3D image segmentation. We evaluate our ACE loss function on four 2D and 3D natural and biomedical image datasets. Our results show that the proposed loss function outperforms other mainstream loss functions on different segmentation networks. Our source code is available at https://github.com/HiLab-git/ACELoss.