Evaluation of continuous approximation functions for the l0-norm for Compressed Sensing

INTRODUCTION: Compressed Sensing (CS) ([1], [2], [3], [4]) allows reconstructing a signal, if it can be represented sparsely in a suitable basis [4], from only a portion of its Fourier coefficients. It was first used by Lustig et al. [5] in MRI, and it has become popular for speeding up the acquisition process. Initially, CS was introduced as an l0-norm minimization [1] which is in practice unsolvable since it is NP-hard. However if the number of acquired samples is increased, this l0-norm minimization is equivalent to an l1norm minimization and therefore faster to compute [1], [2]. It was recently shown in [6] that the CS problem can be solved with less acquired samples than those needed for the l1-norm problem, by minimizing iteratively continuous approximations of the l0-norm. There are a handful continuous functions that can be used to approximate an l0-norm and reconstruct sparse signals [7], but their reconstruction error and convergence properties vary significantly. In this paper we evaluate the performance of four approximation functions using a fixed-point CS solver.