Comparative analysis of sparse signal recovery algorithms based on minimization norms

Abstract

In conventional sensing modality, Nyquist sampling theorem is followed as the minimum sampling rate. However, due to constraints e.g. slow sampling process, limited memory, and sensors cost, in some applications Nyquist sampling rate is difficult to achieve. When sampling rate is less than Nyquist sampling rate, aliasing artifacts occur in the recovered signal. Compressed Sensing (CS) is a modern sampling technique, where signal can be recovered faithfully even from fewer samples if signal/image of interest is sparse, which true as most signals/images are sparse in appropriate domain i.e. Wavelet transform, finite difference. Recovering sparse signal efficiently from compressively sampled data can be most challenging part in CS. The recovery problem is highly ill-posed underdetermined system of linear equations, so additional regularization constraints are required. As there can be infinite many solutions, therefore, finding best solution from few measurements becomes an optimization problem, where cost function is minimized. There are several reconstruction methods that exist in literature. These methods can be classified, based on the norms that are used in minimizing the objective function. This paper presents a comparati ve study of modern sparse signal recovery algorithms using different norms. Sparse signal recovery algorithms presented in this paper are Smoothed l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> magic and mixed l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm based Iterative Shrinkage Algorithms (ISA) e.g. SSF, IRLS and PCD. All algorithms are tested for the recovery of sparse image. The performance measures used for objectively analysing the efficiency of algorithms are mean square error, correlation and computational time.