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Zbl 0903.65001
Stark, Henry; Yang, Yongyi
Vector space projections. A numerical approach to signal and image processing, neural nets, and optics. (English)
[B] Wiley Series in Telecommunications and Signal Processing. Chichester: Wiley. xvi, 408 p. \sterling 55.00 (1998). ISBN 0-471-24140-7/hbk

The book intends to illustrate the method of vector space projections, in particular the method of projections into convex sets. The goal is to present a self-contained approach with a large number of meaningful examples in science and engineering. The reader is assumed to have a first degree in physics, mathematics or engineering.\par First, the basic concepts of vector spaces, inner products and Hilbert spaces are reviewed.\par Chapter 2 introduces the notion of convexity, projectors and projections.\par Vector space projection methods, specifically convex projection methods, always yield a solution consistent with a set of constraints furnished by the user. In Chapter 3, some useful constraints are determined, and a significant number of projections is derived, which are used in signal processing.\par The problem of computation of vector space projections under constraints can be reduced to the solution of a system of linear equations. This method is illustrated in image reconstruction in computerized tomography.\par In Chapter 5, generalized projections are discussed, where the appropriate constraints are not convex. In this case, a restricted type of convergence is possible under certain conditions.\par Chapters 6, 7, 8, 9 are devoted to application of vector space projections in various areas.\par In Chapter 6, the method is applied e.g. to signal reconstruction from non-uniform samples, digital filter design and artifact reduction in image compression.\par Chapter 7 is concerned with projection methods in optics, as e.g. the superresolution problem, the phase retrieval problem, beam forming and design of diffractive optics.\par In Chapter 8, neural nets and pattern recognition systems are considered.\par Finally, the application in image processing (noise-smoothing, image synthesis, high-resolution images, restoration of quantum-limited images) is described in Chapter 9.
[Gerlind Plonka (Duisburg)]

MSC 2000:: *65-01 Textbooks (numerical analysis)
15A03 Vector spaces
46C05 Geometry and topology of inner product spaces
41A29 Approximation with constraints
94A12 Signal theory
46C15 Characterizations of Hilbert spaces
65J05 General theory of numerical methods in abstract spaces
65F10 Iterative methods for linear systems
65D15 Algorithms for functional approximation
68U10 Image processing

Keywords: vector space projections; inner product; Hilbert space; convex projection methods; signal processing; optics; neural nets; image processing; image reconstruction; computerized tomography; signal reconstruction; digital filter design; image compression; pattern recognition

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