Compressed sensing (CS) is an exciting, rapidly growing field that has attracted considerable attention in electrical engineering, applied mathematics, statistics, and computer science. Since its initial introduction several years ago, an avalanche of results have been obtained, both of theoretical and practical nature, and various conferences, workshops, and special sessions have been dedicated to this growing research field. This book provides the first comprehensive introduction to the subject, highlighting recent theoretical advances and a range of applications, as well as outlining numerous remaining research challenges.

CS offers a framework for simultaneous sensing and compression of finite-dimensional vectors, that relies on linear dimensionality reduction. Quite surprisingly, it predicts that sparse high-dimensional signals can be recovered from highly incomplete measurements by using efficient algorithms. To be more specific, let x be a length-n vector. In CS we do not measure x directly, but rather acquire m < n linear measurements of the form y = Ax using an m × n CS matrix A. Ideally, the matrix is designed to reduce the number of measurements as much as possible while allowing for recovery of a wide class of signals from their measurement vectors y. Thus, we would like to choose m ≪ n. However, this renders the matrix A rank-deficient, meaning that it has a nonempty nullspace. This implies that for any particular signal x0, an infinite number of signals x will yield the same measurements y = Ax = Ax0 for the chosen CS matrix.