Locally sparse reconstruction using the $l^{1,\infty}$-norm

Abstract

This paper discusses the incorporation of local sparsity information, e.g. ineach pixel of an image, via minimization of the $\ell^{1,\infty}$-norm. Wediscuss the basic properties of this norm when used as a regularizationfunctional and associated optimization problems, for which we derive equivalentreformulations either more amenable to theory or to numerical computation.Further focus of the analysis is put on the locally 1-sparse case, which iswell motivated by some biomedical imaging applications. Our computational approaches are based on alternating direction methods ofmultipliers (ADMM) and appropriate splittings with augmented Lagrangians. Thoseare tested for a model scenario related to dynamic positron emissiontomography (PET), which is a functional imaging technique in nuclear medicine. The results of this paper provide insight into the potentialimpact of regularization with the $\ell^{1,\infty}$-norm forlocal sparsity in appropriate settings. However, it alsoindicates several shortcomings, possibly related to the non-tightness of thefunctional as a relaxation of the $\ell^{0,\infty}$-norm.