Yin King Chu | Searcharxiv

arXiv2026

A Unified Variational Framework for Deep Weakly Supervised Image Segmentation

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.

cs.CV↗

arXiv2026

A level-wise training scheme for learning neural multigrid smoothers with application to integral equations

Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear systems derived from partial differential equations (PDE) problems, it fails to solve integral equations because its smoothers, which are implemented as conventional relaxation methods, are ineffective in reducing high-frequency components in the errors. We propose a novel neural multigrid scheme where learned neural operators replace classical smoothers. Unlike classical smoothers, these operators are trained offline. Once trained, the neural smoothers generalize to new right-hand-side vectors without retraining, making it an efficient solver. We design level-wise loss functions incorporating spectral filtering to emulate the multigrid frequency decomposition principle, ensuring each operator focuses on solving distinct high-frequency spectral bands. Although we focus on integral equations, the framework is generalizable to all kinds of problems, including PDE problems. Our experiments demonstrate superior efficiency over classical solvers and robust convergence across varying problem sizes and regularization weights.

cs.LG↗