Quality index for detecting reconstruction errors without knowing the signal in l0-norm Compressed Sensing

INTRODUCTION: Compressed Sensing (CS) ([1], [2], [3], [4]) is a recently created algorithm which allows reconstructing a signal from a small portion of its Fourier coefficients if that signal is sparse in a suitable basis. It was first used by Lustig et al. [5] in MRI, and it has become a popular alternative for speeding up the MRI acquisition processes. In practice, CS has been implemented as an l1-norm minimization or a minimization of continuous approximations of an l0-norm [6], [7]. The correct convergence of those minimization approaches is guaranteed only when the number of acquired samples in the Fourier domain is bigger than a certain quantity that depends on the size of the support of the original signal (the number of non-zero coefficients in the sparse domain of the signal). This constitutes a major problem as the support of the signal is not known, and therefore, there is no information to judge whether the reconstructed signal is correct or not. In this article we present a mathematical index that can discriminate correct from erroneous CS reconstructions (without knowing the original signal), and therefore identify the presence of reconstruction artifacts.