# LHMCF-Net: A Learned Hyperbolic Mean Curvature Flow Network for Medical Images Segmentation

arXiv: https://arxiv.org/abs/2608.20942 · HTML: https://mlsanity.com/paper/2608.20942/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 57 references that are in the map.
- First posted on arXiv 21 Aug 2026.

## Builds on (most cited first, 21 of 21)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [DeepLab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs](https://mlsanity.com/paper/1606.00915/) (2016)
- [Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation](https://mlsanity.com/paper/1802.02611/) (2018)
- [Pyramid Scene Parsing Network](https://mlsanity.com/paper/1612.01105/) (2016)
- [ViViT: A Video Vision Transformer](https://mlsanity.com/paper/2103.15691/) (2021)
- [TransUNet: Transformers Make Strong Encoders for Medical Image Segmentation](https://mlsanity.com/paper/2102.04306/) (2021)
- [Attention U-Net: Learning Where to Look for the Pancreas](https://mlsanity.com/paper/1804.03999/) (2018)
- [A Survey on Deep Learning in Medical Image Analysis](https://mlsanity.com/paper/1702.05747/) (2017)
- [Skin Lesion Analysis Toward Melanoma Detection 2018: A Challenge Hosted by the International Skin Imaging Collaboration (ISIC)](https://mlsanity.com/paper/1902.03368/) (2019)
- [Kvasir-SEG: A Segmented Polyp Dataset](https://mlsanity.com/paper/1911.07069/) (2019)
- [UNet++: Redesigning Skip Connections to Exploit Multiscale Features in Image Segmentation](https://mlsanity.com/paper/1912.05074/) (2019)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Polyp-PVT: Polyp Segmentation with Pyramid Vision Transformers](https://mlsanity.com/paper/2108.06932/) (2021)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [FISTA-Net: Learning A Fast Iterative Shrinkage Thresholding Network for Inverse Problems in Imaging](https://mlsanity.com/paper/2008.02683/) (2020)
- [Mumford-Shah Loss Functional for Image Segmentation with Deep Learning](https://mlsanity.com/paper/1904.02872/) (2019)
- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [CNN-based Semantic Segmentation using Level Set Loss](https://mlsanity.com/paper/1910.00950/) (2019)
- [Unfolded proximal neural networks for robust image Gaussian denoising](https://mlsanity.com/paper/2308.03139/) (2023)
- [ℓ1DecNet+: A new architecture framework by ℓ1 decomposition and iteration unfolding for sparse feature segmentation](https://mlsanity.com/paper/2203.02690/) (2022)

## Abstract (arXiv)

Motivated by the classical Chan-Vese model and the ability of deep priors to capture complex spatial structures, we develop a segmentation model that leverages learned hyperbolic mean curvature flow (LHMCF) as a mathematical foundation for integrating feature space data fidelity and deep structural priors within a unified high-dimensional framework. The proposed LHMCF model is governed by a second-order dissipative hyperbolic PDE, where the introduction of a velocity field provides inertia and momentum to the evolving interface. This hyperbolic mechanism enables the contour to bypass noise-induced local minima and propagate coherently through low-contrast or ambiguous regions, addressing limitations inherent to first-order parabolic flows. To solve the continuous LHMCF model, we construct a deep unfolding network, named LHMCF-Net, which maps the iterative numerical procedure of the PDE into a sequence of discrete evolution stages. Each stage corresponds to one physically interpretable update of the underlying dynamical system, allowing the network to inherit the stability and geometric consistency of the PDE while supporting end-to-end optimization. Comprehensive experiments on three publicly available medical segmentation datasets demonstrate that LHMCF-Net achieves superior performance, particularly in challenging scenarios with low contrast and unclear boundaries. These results highlight the effectiveness of embedding hyperbolic geometric evolution into deep unfolding architectures and underscore the potential of physically inspired models for robust medical image segmentation.