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

Recent 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 (GGD's). 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 misspecifications of the prior - a basic concern in image processing applications.