SCANNING THE ISSUE Applications of Sparse Representation and Compressive Sensing

In the past several years, there have been exciting breakthroughs in the study of high-dimensional sparse signals. A sparse signal is a signal that can be represented as a linear combination of relatively few base elements in a basis or an overcomplete dictionary. Much of the excitement centers around the discovery that under surprisingly broad conditions, a sufficiently sparse linear representation can be correctly and efficiently computed by greedy methods and convex optimization (i.e., the ‘ ‘ equivalence), even though this problem is extremely difficultVNP-hard in the general case. Further studies have shown that such high-dimensional sparse signals can be accurately recovered from drastically smaller number of (even randomly selected) linear measurements, hence the catch phrase compressive sensing. If these are not surprising enough, more recently, the same analytical and computational tools have seen similarly remarkable successes in advancing the study of recovering high-dimensional low-rank matrices from highly incomplete, corrupted, and noisy measurements. These results have already caused a small revolution in the community of statistical signal processing as they provide entirely new, or even somewhat paradoxical, perspectives to some of the fundamental principles and doctrines in signal processing such as the sampling bounds and the choice of bases for signal representation and reconstruction. A recent IEEE SIGNAL PROCESSING MAGAZINE special issue on compressive sampling has captured some of the most recent and exciting developments in this field. We, the guest editors of this special issue, strongly believe that these new results and the general mathematical principles behind them are of great interest to scientific and engineering communities far beyond signal processing. These new results and revelations have forever changed our perspective and enhanced our ability in acquiring, processing, and analyzing massive high-dimensional data, regardless of their physical nature. Therefore, the theme of this new Sparse representation and compressive sensing establishes a more rigorous mathematical framework for studying high-dimensional data and ways to uncover the structures of the data, giving rise to a large repertoire of efficient algorithms.