By Yonina C. Eldar, Gitta Kutyniok
Compressed sensing is an exhilarating, quickly transforming into box, attracting substantial cognizance in electric engineering, utilized arithmetic, statistics and desktop technological know-how. This ebook offers the 1st specific creation to the topic, highlighting contemporary theoretical advances and a number of functions, in addition to outlining a variety of ultimate examine demanding situations. After a radical overview of the elemental conception, many state-of-the-art ideas are provided, together with complicated sign modeling, sub-Nyquist sampling of analog indications, non-asymptotic research of random matrices, adaptive sensing, grasping algorithms and use of graphical types. All chapters are written via top researchers within the box, and constant sort and notation are applied all through. Key historical past info and transparent definitions make this a great source for researchers, graduate scholars and practitioners eager to sign up for this intriguing learn region. it could additionally function a supplementary textbook for classes on computing device imaginative and prescient, coding thought, sign processing, picture processing and algorithms for effective information processing.
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Additional info for Compressed Sensing: Theory and Applications
C. Eldar, and G. Kutyniok signal x ∈ Rn , we consider measurement systems that acquire m linear measurements. 4) where A is an m × n matrix and y ∈ Rm . , it maps Rn , where n is generally large, into Rm , where m is typically much smaller than n. Note that in the standard CS framework we assume that the measurements are non-adaptive, meaning that the rows of A are fixed in advance and do not depend on the previously acquired measurements. In certain settings adaptive measurement schemes can lead to significant performance gains.
4, in many cases the introduction of Φ does not significantly complicate the construction of matrices A such that A will satisfy the desired properties. Thus, for the remainder of this chapter we will restrict our attention to the case where Φ = I. It is important to note, however, that this restriction does impose certain limits in our analysis when Φ is a general dictionary and not an orthonormal basis. For example, in this case x − x 2 = Φc − Φc 2 = c − c 2 , and thus a bound on c − c 2 cannot directly be translated into a bound on x − x 2 , which is often the metric of interest.
If we wish to be able to recover all sparse signals x from the measurements Ax, then it is immediately clear that for any pair of distinct vectors x, x ∈ Σk , we must have Ax = Ax , since otherwise it would be impossible to distinguish x from x based solely on the measurements y. More formally, by observing that if Ax = Ax then A(x−x ) = 0 with x − x ∈ Σ2k , we see that A uniquely represents all x ∈ Σk if and only if N (A) contains no vectors in Σ2k . While there are many equivalent ways of characterizing this property, one of the most common is known as the spark .