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

Abstract

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 coe cients and heavy{tailed priors such as Generalized Gaussian distributions (GGDs). In this paper, we investigate various connections between shrinkage methods and 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 wavelet{based MDL estimators. 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 misspeci cations of the prior { a basic concern in image processing applications.

Natural image modeling using complex wavelets

“…5): they relate to the full coefficient distribution instead of its order n moments. Then, the marginal distribution of a wavelet coefficient ξ k,l (where k is the subband index and l a spatial index) is modeled as a Generalized Gaussian (GG) distribution, 14,15 which has the remarkable property to be scale invariant:…”

Statistical Modeling of Photographic Images

“…For example, when applied to the problem of compression, the entropy of the distributions described above is significantly less than that of a Gaussian with the same variance, and this leads directly to gains in coding efficiency. In denoising, the use of this model as a prior density for images yields to significant improvements over the Gaussian model [e.g., 48,11,2,34,47]. Consider again the problem of removing additive Gaussian white noise from an image.…”

“…4. For natural images, these histograms are surprisingly well described by a two-parameter generalized Gaussian (also known as a stretched, or generalized exponential) distribution [e.g., 31,47,34]:…”

Image denoising based on the edge-process model

“…GGd and Gaussian distribution correspondingly) [15]. The main drawback of these models is their inability to capture local statistics that play a crucial role in denoising applications.…”