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An introduction to compressed sensing
Vidyasagar M., SIAM, Philadelphia, PA, 2020. 341 pp. Type: Book (978-1-611976-11-3)
Date Reviewed: Feb 4 2021

Compressed sensing has emerged as a key technology in the signal processing area. This disruptive technique has found applications in many real-life problems. We know from the classical Nyquist-Shannon theorem that a signal can be recovered perfectly from a set of uniformly spaced samples, taken at a rate of twice the highest frequency present. Compressed sensing allows us to recover the sparse signal perfectly when sampled below the Nyquist rate. In other words: it’s all about recovering the low-complexity but high-dimensional object from a limited number of measurements. The current book gives a mathematical introduction to the subject of compressed sensing.

The book’s ten chapters, including a bibliography, index, and preface, contain all the basic mathematical techniques of compressed sensing. Chapter 1 covers basic mathematical preliminaries such as norms, singular value decomposition (SVD), convexity, and probability. Chapter 2 covers the problem formulation, and provides some historical background. Null space-based conditions and the restricted isometry property are discussed in chapter 3. Some concepts from graph theory, such as Ramanujan graphs, are covered in chapter 4. Deterministic and probabilistic constructions of measurement matrices are given in chapters 5 and 6, respectively. Matrix recovery and matrix completion methods are discussed in chapter 7. Chapter 8 covers approaches for vector recovery. Optimal algorithms are discussed in chapter 9. Finally, case studies on vector recovery and matrix completion are given in chapter 10.

After reading the book, I can say it’s written for a mathematically mature audience; for example, PhD students and experienced scientists and engineers will enjoy reading it. Nevertheless, it can be used as a textbook for an elective course in the area of computer science and communications (especially signal processing) for advanced undergraduate and graduate students. I recommend this introductory book on compressed sensing to mathematicians and engineers.

Reviewer:  Manish Gupta Review #: CR147179 (2104-0070)
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