A General Framework for Sparsity-Based Denoising and Inversion

Abstract

Estimating a reliable and stable solution to many problems in signal processing and imaging is based on sparse regularizations, where the true solution is known to have a sparse representation in a given basis. Using different approaches, a large variety of regularization terms have been proposed in literature. While it seems that all of them have so much in common, a general potential function which fits most of them is still missing. In this paper, in order to propose an efficient reconstruction method based on a variational approach and involving a general regularization term (including most of the known potential functions, convex and nonconvex), we deal with i) the definition of such a general potential function, ii) the properties of the associated “proximity operator” (such as the existence of a discontinuity), and iii) the design of an approximate solution of the general “proximity operator” in a simple closed form. We also demonstrate that a special case of the resulting “proximity operator” is a set of shrinkage functions which continuously interpolate between the soft-thresholding and hard-thresholding. Computational experiments show that the proposed general regularization term performs better than <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> -penalties for sparse approximation problems. Some numerical experiments are included to illustrate the effectiveness of the presented new potential function.