# A Unified Variational Framework for Deep Weakly Supervised Image Segmentation

arXiv: https://arxiv.org/abs/2607.19669 · HTML: https://mlsanity.com/paper/2607.19669/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 50 references that are in the map.
- First posted on arXiv 22 Jul 2026.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [V-Net: Fully Convolutional Neural Networks for Volumetric Medical Image Segmentation](https://mlsanity.com/paper/1606.04797/) (2016)
- [Segmenter: Transformer for Semantic Segmentation](https://mlsanity.com/paper/2105.05633/) (2021)
- [ScribbleSup: Scribble-Supervised Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1604.05144/) (2016)
- [Learning Pixel-level Semantic Affinity with Image-level Supervision for Weakly Supervised Semantic Segmentation](https://mlsanity.com/paper/1803.10464/) (2018)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [On Regularized Losses for Weakly-supervised CNN Segmentation](https://mlsanity.com/paper/1803.09569/) (2018)
- [Normalized Cut Loss for Weakly-supervised CNN Segmentation](https://mlsanity.com/paper/1804.01346/) (2018)
- [Constrained Convolutional Neural Networks for Weakly Supervised Segmentation](https://mlsanity.com/paper/1506.03648/) (2015)
- [Tree Energy Loss: Towards Sparsely Annotated Semantic Segmentation](https://mlsanity.com/paper/2203.10739/) (2022)
- [Geometry Constrained Weakly Supervised Object Localization](https://mlsanity.com/paper/2007.09727/) (2020)
- [Mumford-Shah Loss Functional for Image Segmentation with Deep Learning](https://mlsanity.com/paper/1904.02872/) (2019)
- [Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-shape Priori for Image Segmentation](https://mlsanity.com/paper/2002.03989/) (2020)
- [PottsMGNet: A Mathematical Explanation of Encoder-Decoder Based Neural Networks](https://mlsanity.com/paper/2307.09039/) (2023)
- [Weakly supervised segmentation with point annotations for histopathology images via contrast-based variational model](https://mlsanity.com/paper/2304.03572/) (2023)
- [An efficient iterative thresholding method for image segmentation](https://mlsanity.com/paper/1608.01431/) (2016)
- [Deep Convolutional Neural Networks Meet Variational Shape Compactness Priors for Image Segmentation](https://mlsanity.com/paper/2406.19400/) (2024)

## Abstract (arXiv)

We propose a unified variational framework for image segmentation under sparse pixel-level supervision. Our method is based on a simplex-constrained Potts model with a smooth perimeter regularizer, yielding a convex, smooth energy functional that can be used as a training loss in weakly supervised deep learning paradigms or optimized efficiently using iterative methods. Sparse labels are incorporated into the data fidelity term by constructing a fuzzy membership function via a function extension problem in a Reproducing Kernel Hilbert Space (RKHS), which can effectively capture inhomogeneous intensity statistics. The derived discrete loss for training standard networks demonstrates robustness and consistent improvements over non-training and partial cross-entropy (PCE) baselines in experiments, achieving comparable performance without requiring ground-truth segmentation images.