A Mathematical Introduction to Compressive Sensing by Simon Foucart

By Simon Foucart

At the intersection of arithmetic, engineering, and machine technology sits the thriving box of compressive sensing. in response to the idea that facts acquisition and compression could be played concurrently, compressive sensing unearths functions in imaging, sign processing, and plenty of different domain names. within the parts of utilized arithmetic, electric engineering, and theoretical laptop technological know-how, an explosion of analysis job has already the theoretical effects that highlighted the potency of the fundamental ideas. The dependent rules in the back of those ideas also are of self reliant curiosity to natural mathematicians.

A Mathematical advent to Compressive Sensing supplies an in depth account of the center conception upon which the sector is construct. With merely reasonable must haves, it's a very good 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 gather a cautious knowing of the topic. A Mathematical advent to Compressive Sensing makes use of a mathematical viewpoint to give the center of the idea underlying compressive sensing.

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Chapter 6 starts with basic results on the restricted isometry constants. For instance, there is the relation δ2 = μ with the coherence when the columns of A are 2 -normalized. In this sense, restricted isometry constants generalize the coherence by considering all s-tuples rather than all pairs of columns. Other relations include the simple (and quite pessimistic) bound δs ≤ (s − 1)μ, which can be derived directly from Gershgorin’s disk theorem. We then turn to the analysis of the various recovery algorithms based on the restricted isometry property of A.

BN2 ) of different nature (for instance, sinusoids and spikes), then the situation changes. 2 Applications, Motivations, and Extensions 19 where the matrix A ∈ Cm×(N1 +N2 ) has columns a1 , . . , aN1 , b1 , . . , bN2 and the vector x = [x1,1 , . . , x1,N1 , x2,1 , . . , x2,N2 ] is sparse. The compressive sensing methodology then allows one—under certain conditions—to determine N1 the coefficient vector x, hence to derive the two components y1 = j=1 x1,j aj and y2 = N2 j=1 x2,j bj . Error Correction In every realistic data transmission device, pieces of data are occasionally corrupted.

In fact, we will see in Chap. 12 that choosing the sampling points t1 , . . , tm independently and uniformly at random in [0, 1] allows one to reconstruct f with high probability from its m samples f (t1 ), . . , f (tm ) provided that m ≥ Cs ln(N ). Thus, few samples suffice if s is small. An illustrating example was already displayed in Figs. 3. 2 Applications, Motivations, and Extensions 17 Sparse Approximation Compressive sensing builds on the empirical observation that many types of signals can be approximated by sparse ones.

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