A Survey of Compressed Sensing

Compressed sensing was introduced some ten years ago as an effective way of acquiring signals, which possess a sparse or nearly sparse representation in a suitable basis or dictionary. Due to its solid mathematical backgrounds, it quickly attracted the attention of mathematicians from several different areas, so that the most important aspects of the theory are nowadays very well understood. In recent years, its applications started to spread out through applied mathematics, signal processing, and electrical engineering. The aim of this chapter is to provide an introduction into the basic concepts of compressed sensing. In the first part of this chapter, we present the basic mathematical concepts of compressed sensing, including the Null Space Property, Restricted Isometry Property, their connection to basis pursuit and sparse recovery, and construction of matrices with small restricted isometry constants. This presentation is easily accessible, largely self-contained, and includes proofs of the most important theorems. The second part gives an overview of the most important extensions of these ideas, including recovery of vectors with sparse representation in frames and dictionaries, discussion of (in)coherence and its implications for compressed sensing, and presentation of other algorithms of sparse recovery.