By Simon Foucart

ISBN-10: 0817649476

ISBN-13: 9780817649470

ISBN-10: 0817649484

ISBN-13: 9780817649487

At the intersection of arithmetic, engineering, and machine technology sits the thriving box of compressive sensing. according to the basis that facts acquisition and compression may be played at the same time, compressive sensing reveals functions in imaging, sign processing, and lots of different domain names. within the components of utilized arithmetic, electric engineering, and theoretical desktop technology, an explosion of study task has already the theoretical effects that highlighted the potency of the fundamental ideas. The dependent rules at the back of those ideas also are of self reliant curiosity to natural mathematicians.

*A Mathematical advent to Compressive Sensing* offers a close account of the center thought upon which the sector is construct. With purely reasonable necessities, it's a great textbook for graduate classes in arithmetic, engineering, and laptop technological know-how. It additionally serves as a competent source for practitioners and researchers in those disciplines who are looking to collect a cautious figuring out of the topic. *A Mathematical creation to Compressive Sensing* makes use of a mathematical viewpoint to provide the middle of the speculation underlying compressive sensing.

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**Extra resources for A Mathematical Introduction to Compressive Sensing**

**Example text**

13 play a key role in some of the constructions [41]. Quite remarkably, sublinear algorithms are also available for sparse Fourier transforms [223, 261, 262, 287, 288, 519]. Applications of Compressive Sensing. We next provide comments and references on the applications and motivations described in Sect. 2. Single-pixel camera. The single-pixel camera was developed by Baraniuk and coworkers [174] as an elegant proof of concept that the ideas of compressive sensing can be implemented in hardware.

An estimate for x − x often yields an estimate for y − y = A(x − x ) , but the converse is not generally true. Finally, we briefly describe some signal and image processing applications of sparse approximation. ˆ = Aˆ • Compression. Suppose that we have found a sparse approximation y x of ˆ . Then storing y ˆ amounts to storing only the a signal y with a sparse vector x ˆ . Since x ˆ is sparse, significantly less memory is required nonzero coefficients of x than for storing the entries of the original signal y.

2 Applications, Motivations, and Extensions 21 y = f (t) + e, where e is random noise. The task is to learn the function f based on training samples (tj , yj ). Without further hypotheses on f , this is an impossible task. Therefore, we assume that f has a sparse expansion in a given dictionary of functions ψ1 , . . , that f is written as N f (t) = x ψ (t), =1 where x is a sparse vector. Introducing the matrix A ∈ Rm×N with entries Aj,k = ψk (tj ), we arrive at the model y = Ax + e, and the task is to estimate the sparse coefficient vector x.

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