Image Denoising with Shrinkage and Redundant Representations

Shrinkage is a well known and appealing denoising technique. The use of shrinkage is known to be optimal for Gaussian white noise, provided that the sparsity on the signal’s representation is enforced using a unitary transform. Still, shrinkage is also practiced successfully with nonunitary, and even redundant representations. In this paper we shed some light on this behavior. We show that simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising (BPDN) problem. Thus, this work leads to a novel iterative shrinkage algorithm that can be considered as an effective pursuit method. We demonstrate this algorithm, both on synthetic data, and for the image denoising problem, where we learn the image prior parameters directly from the given image. The results in both cases are superior to several popular alternatives.