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This paper introduces a unified variational framework for image segmentation that operates effectively under sparse pixel-level supervision by leveraging a simplex-constrained Potts model with a smooth perimeter regularizer. By incorporating sparse labels through a fuzzy membership function derived from a Reproducing Kernel Hilbert Space, the method captures complex intensity variations and leads to a convex energy functional suitable for training deep learning models. Experimental results show that this approach consistently outperforms traditional non-training and partial cross-entropy baselines, achieving performance comparable to fully supervised methods without the need for ground-truth segmentation images.
Sparse pixel-level supervision can yield segmentation results on par with fully supervised methods, revolutionizing the efficiency of training deep learning models.
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.
