Random Fields Estimation Theory: (48 Pitman Monographs & Surveys in Pure & Applied Mathematics.)

This book presents the analytical theory of random fields estimation by the criterion of minimum of variance. The theory generalizes the classical Wiener theory to the case of random functions of several variables, that is, random fields. There are many applied problems which can be formulated as random fields estimation problems. Such problems arise in optics and TV signal processing, geophysics, underwater acoustics, image processing in photography to mention a few. The author developed the theory presented in this monograph in a series of papers which are mentioned in the bibliographical notes. Analytical formulas for the optimal filters are given. These filters are determined generically by distributional kernels. The singularities of the optimal filters are studied in detail and the results are in a final form for the class of random fields introduced in this monograph. A numerical method is developed for solving the basic integral equation of the estimation theory in distributions. A number of applied problems is discussed and a considerable amount of the the auxiliary material from functional analysis and probability theory has to be included. This material is given partly as a reference material, without proofs, and partly with proofs, especially in the cases when this material cannot be found in the literature. The book should be of particlar interest to engineers and geophysicists.

Table of Contents:
Part 1 Formulation of basic results: statements of the problem; formulation of results (multidimensional case) and (one-dimensional case); examples of kernels of class R and solutions to the basic equation; formula for the error of optimal estimate. Part 2 Numerical solution of the basic integral equation in distributions: basic ideas; theoretical approaches; multidimensional equation; numerical solution based on the approximation of the kernel; asymptotic behaviour of the optimal filter as the white noise component goes to zero. Part 3 Proofs: proof of theorem II.2.1; proof of theorem II.2.2; proof of theorems II.3.1. and II.3.2. Part 4 Applications; what is the optimal size of the domain on which the data are to be collected?; discrimination of random fields against the noisy background; quasioptimal estimates of derivatives of random functions; stable summation of orthogonal series and integrals with randomly perturbed coefficients; resolution ability of linear systems; ill-posed problems and estimation theory; nonlinear (polynomial) estimates. Part 5 Auxiliary results: Sobolev spaces and distributions; Eigenfunction expansions for elliptic selfadjoint operators; asymptotics of the spectrum of linear operators; elements of probability theory. Appendix: estimation and scattering theory.