Gini Index as Sparsity Measure for Signal Reconstruction from Compressive Samples

Abstract

Sparsity is a fundamental concept in compressive sampling of signals/images, which is commonly measured using the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm, even though, in practice, the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> or the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ( 0 <; <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <; 1) (pseudo-) norm is preferred. In this paper, we explore the use of the Gini index (GI), of a discrete signal, as a more effective measure of its sparsity for a significantly improved performance in its reconstruction from compressive samples. We also successfully incorporate the GI into a stochastic optimization algorithm for signal reconstruction from compressive samples and illustrate our approach with both synthetic and real signals/images.