1. INTRODUCTION
In the classical atomic decomposition problem [1], we have a signal whose samples are collected in the signal vector and we would like to represent it as a linear combination of signal vectors . After [2], the vectors are called atoms and they collectively form a dictionary over which the signal is to be decomposed. We may write {\bf s}=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}=\Phi\alpha, \eqno{\hbox{(1)}} where is the dictionary (matrix) and is the vector of coefficients. A dictionary with is called overcomplete. Although, is sufficient to obtain such a decomposition (like what is done in Discrete Fourier Transform), using overcomplete dictionaries has a lot of advantages in many diverse applications (refer for example to [3] and the references in it). Note that for the overcomplete case, the representation is not unique, but all these applications need a sparse representation, that is, the signal s should be represented as a linear combination of as small as possible number of atoms of the dictionary. It has been shown [4], [5] that with some mild conditions on the dictionary matrix, if there is a sparse representation with at most non-zero coefficients, then this representation is unique. The main approaches for finding this sparse solution include Matching Pursuit (MP) [2], [6], FOCUSS [5], Basis Pursuit (BP) [1], and Smoothed (SLO) [7].