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A mathematical introduction to compressive sensing
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| Recent years have seen an explosion of research activities in a fascinating area called compressed sensing, compressive sensing, or compressive sampling. A Google Scholar search for articles containing one of these three terms in their title returned about 4,400 hits at the time this preface was written. The area of compressive sens- ing, at the intersection of mathematics, electrical engineering, computer science, and physics, takes its name from the premise that data acquisition and compression can be performed simultaneously. This is possible because many real-world signals are sparse, and even though they are acquired with seemingly too few measurements, exploiting sparsity enables one to solve the resulting underdetermined systems of linear equations. The reconstruction of sparse signals is not only feasible in theory, but efficient algorithms also exist to perform the reconstruction in practice. Moreover, involving randomness in the acquisition step enables one to utilize the minimal number of measurements. These realizations, together with their potential applications, have triggered the interest of the scientific community since around 2004. Some of the ingredients are of course much older than the advent of compressive sensing itself, and the underlying theory builds on various branches of mathematics. These branches include linear algebra, approximation theory, convex analysis, optimization, probability theory (in particular, random matrices), Banach space geometry, harmonic analysis, and graph theory. This book is a detailed and self-contained introduction to the rich and elegant mathematical theory of compressive sensing. It presents all the necessary background material without assuming any special prior knowledge—just basic analysis, linear algebra, and probability theory. |
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