Journal
IEEE TRANSACTIONS ON SIGNAL PROCESSING
Volume 59, Issue 11, Pages 5202-5211Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TSP.2011.2164074
Keywords
Potential function; proximity operator; regularization; sparse approximation
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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 l(p)-penalties for sparse approximation problems. Some numerical experiments are included to illustrate the effectiveness of the presented new potential function.
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