L1-L2 Optimization in Signal and Image Processing

Abstract

Sparse, redundant representations offer a powerful emerging model for signals. This model approximates a data source as a linear combination of few atoms from a prespecified and over-complete dictionary. Often such models are fit to data by solving mixed ¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -¿ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> convex optimization problems. Iterative-shrinkage algorithms constitute a new family of highly effective numerical methods for handling these problems, surpassing traditional optimization techniques. In this article, we give a broad view of this group of methods, derive some of them, show accelerations based on the sequential subspace optimization (SESOP), fast iterative soft-thresholding algorithm (FISTA) and the conjugate gradient (CG) method, present a comparative performance, and discuss their potential in various applications, such as compressed sensing, computed tomography, and deblurring.