By Yonina C. Eldar, Gitta Kutyniok
Compressed sensing is an exhilarating, quickly becoming box, attracting significant cognizance in electric engineering, utilized arithmetic, data and laptop technological know-how. This booklet offers the 1st distinctive advent to the topic, highlighting fresh theoretical advances and more than a few purposes, in addition to outlining quite a few closing learn demanding situations. After a radical overview of the fundamental idea, many state of the art strategies are awarded, together with complex sign modeling, sub-Nyquist sampling of analog signs, non-asymptotic research of random matrices, adaptive sensing, grasping algorithms and use of graphical versions. All chapters are written by way of major researchers within the box, and constant sort and notation are applied all through. Key heritage info and transparent definitions make this a fantastic source for researchers, graduate scholars and practitioners eager to subscribe to this intriguing examine zone. it may possibly additionally function a supplementary textbook for classes on machine imaginative and prescient, coding thought, sign processing, photograph processing and algorithms for effective information processing.
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Additional info for Compressed Sensing : Theory and Applications
If we assume that the columns of A have unit norm, then each coefficient of AT e is a Gaussian random variable with mean zero and variance σ 2 . 5) of Chapter 5), we have that P AT e i ≥ tσ ≤ exp −t2 /2 for i = 1, 2, . . , n. Thus, using the union bound over the bounds for different i, we obtain P AT e ∞ ≥2 log nσ ≤ n exp (−2 log n) = 1 . 1 of . 2 √ Suppose that A has unit-norm columns and satisfies the RIP of order 2k with δ2k < 2 − 1. d. N (0, σ 2 ). 12) obeys x−x √ 2≤4 2 √ 1 + δ2k √ 1 − (1 + 2)δ2k k log nσ with probability at least 1 − n1 .
65 of Chapter 5 states that if a matrix A is chosen according to a sub-gaussian distribution 2 with m = O k log(n/k)/δ2k , then A will satisfy the RIP of order 2k with probability at 2 least 1 − 2 exp(−c1 δ2k m). Results of this kind open the door to slightly weaker results that hold only with high probability. Even within the class of probabilistic results, there are two distinct flavors. The typical approach is to combine a probabilistic construction of a matrix that will satisfy the RIP with high probability with the previous results in this chapter.
The distinction is essentially whether or not we need to draw a new random A for each signal x. 13 may be somewhat pessimistic, exhibited by the following result. 14 Let x ∈ Rn be fixed. Set δ2k < 2 − 1. Suppose that A is an m × n 2 sub-gaussian random matrix with m = O k log(n/k)/δ2k . Suppose we obtain measurements of the form y = Ax. Set = 2σk (x)2 . 12) obeys x ˆ−x 2≤ √ √ 8 1 + δ2k − (1 + 2)δ2k √ σk (x)2 . 1 − (1 + 2)δ2k Proof. 65 of Chapter 5 we have that 2 m). Next, let A will satisfy the RIP of order 2k with probability at least 1 − 2 exp(−c1 δ2k Λ denote the index set corresponding to the k entries of x with largest magnitude and write x = xΛ + xΛc .