Provably Convergent Low-Rank Grayscale Image Fusion via Adaptive Weighted Nuclear Norm Minimization

We propose a fully unsupervised, convex, and globally convergent framework for multimodal grayscale image fusion based on adaptive weighted nuclear norm minimization (WNNM). By formulating the fusion task as a low-rank matrix recovery problem with data-driven singular value reweighting, the method automatically preserves salient structural features from both source images while effectively suppressing noise and redundancy. We establish global convergence of the proximal gradient algorithm with a Q-linear rate, derive a new perturbation error bound based on weighted Eckart–Young–Mirsky theory, and accelerate convergence using FISTA. Extensive experiments on the TNO, RoadScene, and Harvard Medical imaging datasets, comprising over 100 image pairs, demonstrate that the proposed approach achieves state-of-the-art performance in terms of entropy (EN), edge-based quality index (QAB/F), visual information fidelity (VIF), spatial frequency (SF), and subjective visual quality. The method consistently outperforms classical matrix-based approaches and performs competitively with recent supervised deep learning methods while requiring neither training data nor GPU acceleration.