Vector space projections. A numerical approach to signal and image processing, neural nets, and optics.

*(English)*Zbl 0903.65001
Wiley Series in Telecommunications and Signal Processing. Chichester: Wiley. xvi, 408 p. (1998).

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.

First, the basic concepts of vector spaces, inner products and Hilbert spaces are reviewed.

Chapter 2 introduces the notion of convexity, projectors and projections.

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.

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.

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.

Chapters 6, 7, 8, 9 are devoted to application of vector space projections in various areas.

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.

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.

In Chapter 8, neural nets and pattern recognition systems are considered.

Finally, the application in image processing (noise-smoothing, image synthesis, high-resolution images, restoration of quantum-limited images) is described in Chapter 9.

First, the basic concepts of vector spaces, inner products and Hilbert spaces are reviewed.

Chapter 2 introduces the notion of convexity, projectors and projections.

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.

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.

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.

Chapters 6, 7, 8, 9 are devoted to application of vector space projections in various areas.

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.

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.

In Chapter 8, neural nets and pattern recognition systems are considered.

Finally, the application in image processing (noise-smoothing, image synthesis, high-resolution images, restoration of quantum-limited images) is described in Chapter 9.

Reviewer: Gerlind Plonka (Duisburg)

##### MSC:

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

15A03 | Vector spaces, linear dependence, rank, lineability |

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

41A29 | Approximation with constraints |

94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |

46C15 | Characterizations of Hilbert spaces |

65J05 | General theory of numerical analysis in abstract spaces |

65F10 | Iterative numerical methods for linear systems |

65D15 | Algorithms for approximation of functions |

68U10 | Computing methodologies for image processing |