Analysis of multiresolution image denoising schemes using generalized Gaussian and complexity priors

TLDR

Universal wavelet shrinkage achieves near‑ideal performance asymptotically, yet in practice Bayesian estimators with heavy‑tailed priors such as generalized Gaussian distributions outperform universal thresholding. The study examines how shrinkage methods relate to MAP estimation with heavy‑tailed priors, establishes a sparsity condition for MAP estimates, and introduces a new family of complexity priors based on Rissanen’s universal prior on integers. Analytical expressions for shrinkage rules derived from generalized Gaussian and the new complexity priors are developed, providing explicit MAP estimators. One estimator from this class surpasses conventional MDL‑based estimators, and the analysis shows that universal hard thresholding, MAP with a very heavy‑tailed GGD, and MDL with the new complexity prior are equivalent, with experiments confirming robustness to prior misspecification.

Abstract

Research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near-ideal estimation performance in various asymptotic frameworks. However, image processing practice has shown that universal thresholding methods are outperformed by simple Bayesian estimators assuming independent wavelet coefficients and heavy-tailed priors such as generalized Gaussian distributions (GGDs). In this paper, we investigate various connections between shrinkage methods and maximum a posteriori (MAP) estimation using such priors. In particular, we state a simple condition under which MAP estimates are sparse. We also introduce a new family of complexity priors based upon Rissanen's universal prior on integers. One particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length (MDL) principle. We develop analytical expressions for the shrinkage rules implied by GGD and complexity priors. This allows us to show the equivalence between universal hard thresholding, MAP estimation using a very heavy-tailed GGD, and MDL estimation using one of the new complexity priors. Theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis-specifications of the prior-a basic concern in image processing applications.