Data Separation by Sparse Representations

Recently, sparsity has become a key concept in various areas of applied mathematics, computer science, and electrical engineering. One application of this novel methodology is the separation of data, which is composed of two (or more) morphologically distinct constituents. The key idea is to carefully select representation systems each providing sparse approximations of one of the components. Then the sparsest coefficient vector representing the data within the composed - and therefore highly redundant - representation system is computed by $\ell_1$ minimization or thresholding. This automatically enforces separation. This paper shall serve as an introduction to and a survey about this exciting area of research as well as a reference for the state-of-the-art of this research field. It will appear as a chapter in a book on "Compressed Sensing: Theory and Applications" edited by Yonina Eldar and Gitta Kutyniok.