Closed-Form Nonlocal Shrinkage for Multiplicative Image Denoising and SAR Despeckling

Multiplicative noise poses a challenge in coherent and signal-dependent imaging owing to its intensity-dependent variance and frequently non-Gaussian distribution. We propose a deterministic nonlocal estimator that combines a logarithmic Yeo--Johnson transformation, patch grouping, an adaptive singular basis, and sparse shrinkage. The orthonormal group dictionary makes the weighted Lasso separable and yields an exact coefficient-wise soft-threshold solution. This solution replaces the iterative inner solver and expresses patch reliability and atom importance through a single threshold field. Since the dictionary is estimated from the noisy group, we introduce a random-matrix correction governed by the group aspect ratio γ=p2/K. The correction links patch size, group size, and shrinkage strength. Experiments cover gamma-corrupted images from three standard benchmarks and real synthetic aperture radar (SAR) imagery from five sensors. The method gives the best result in 18 of 24 PSNR/SSIM comparisons with twelve published methods and the lowest mean ratio-image deviation across six real SAR configurations. These results support geometry-calibrated nonlocal modeling for structure-preserving image restoration, with SAR despeckling serving as a demanding application. Code is available hrefhttps://github.com/Teriri1999/Geometry-Calibrated-Closed-Form-Shrinkage-for-SAR-Despecklinghere.