# A Unified Variational Framework for Deep Weakly Supervised Image Segmentation

> Source: <https://arxiv.org/abs/2607.19669>
> Published: 2026-07-23 04:00:00+00:00

arXiv:2607.19669v1 Announce Type: new
Abstract: 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.