A Unified Variational Framework for Deep Weakly Supervised Image Segmentation

2026-07-21

AI Summary

This work addresses the challenge of image segmentation with only sparse pixel-level annotations by proposing a unified variational framework. The approach extends sparse labels into fuzzy membership functions within a reproducing kernel Hilbert space (RKHS) to effectively model non-uniform intensity statistics. It combines a simplex-constrained Potts model with a smooth perimeter regularizer to formulate a convex and differentiable energy functional. This framework supports both iterative optimization and integration into weakly supervised deep learning pipelines. Experimental results demonstrate that the proposed method significantly outperforms conventional non-learning approaches and cross-entropy-based baselines, achieving segmentation performance on par with fully supervised methods when incorporated into standard neural architectures.

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.

Problem

• weakly supervised learning
• image segmentation
• sparse annotation
• pixel-level supervision
• variational framework

Innovation

• variational framework
• weakly supervised segmentation
• Potts model
• RKHS
• sparse supervision