Image Restoration with Multiplicative Noise: A Hybrid Geometric Model and Its Alternating Minimization Algorithm

Abstract.

We propose a variational framework for joint removal of blur and multiplicative noise. While geometric priors effectively capture intrinsic image structure, relying on a single prior often introduces characteristic artifacts. Motivated by the geometric relationship between mean curvature and surface area, and their different differential orders, we introduce a hybrid regularization that combines these two complementary quantities to mitigate their respective drawbacks. A diffusion-based analysis further reveals that the proposed regularizer provides a balanced control over both area and curvature, thereby preserving the advantages of each component. The resulting nonconvex, nonsmooth model is solved by an alternating minimization algorithm with an appropriate variable splitting, which decomposes the original problem into a sequence of tractable subproblems-each admitting either a closed-form update or an efficient numerical solution. We provide a rigorous analysis showing global convergence of the generated iterates to a critical point under mild conditions. Extensive experiments demonstrate the effectiveness of the proposed model in jointly removing blur and multiplicative noise, as well as the efficiency of the optimization algorithm.