An eigenstructure approach for the retrieval of cylindrical harmonics (original) (raw)
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Linear statistical problems for stationary isotropic random fields on a sphere. I
Theory of Probability and Mathematical Statistics, No. 18,1979, 1979
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This paper presents a maximum-likelihood solution to the general problem of tting a parametric model to observations from a single realization of a 2-D homogeneous random eld with mixed spectral distribution. On the basis of a 2-D Wold-like decomposition, the eld is represented as a sum of mutually orthogonal components of three types: purelyindeterministic, harmonic, and evanescent. The suggested algorithm involves a two-stage procedure. In the rst stage we obtain a suboptimal initial estimate for the parameters of the spectral support of the evanescent and harmonic components. In the second stage we re ne these initial estimates by iterative maximization of the conditional likelihood of the observed data, which is expressed as a function of only the parameters of the spectral supports of the evanescent and harmonic components. The solution for the unknown spectral supports of the harmonic and evanescent components reduces the problem of solving for the other unknown parameters of the eld to linear least squares. The Cramer-Rao lower bound on the accuracy of jointly estimating the parameters of the di erent components is derived, and it is shown that the bounds on the purely-indeterministic and deterministic components are decoupled. Numerical evaluation of the bounds provides some insight into the e ects of various parameters on the achievable estimation accuracy. The performance of the Maximum-Likelihood algorithm is illustrated by Monte-Carlo simulations and is compared with the Cramer-Rao bound. Index terms: ML estimation of 2-D random elds, 2-D Wold decomposition, 2-D mixed spectral distributions, purely-indeterministic elds, harmonic elds, evanescent elds, Cramer-Rao bound.
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Theory of Probability and Mathematical Statistics, No. 53, 1996, 1996
A b s t r a c t . This article is the second part of [1]. The problem o f the least (in mean square) linear estimate o f a functional of the unknown values of a time-homogeneous random field £ (t ,x ) isotropic on a sphere Sn using observations of the field £ (t,x ) for t < 0, x 6 Sn is considered. The least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional are determined for some special classes of spectral densities.
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This paper considers the application of Kalman estimation theory to the problem of estimating two-dimensional isotropic random fields, whose equations are expressed in terms of the Laplacian, given some noisy observations on a finite disk. It is shown that this problem is equivalent to that of solving a countably infinite number of one-dimensional estimation problems. Markovian models for the one-dimensional processes are developed and the associated Kalman filters are shown to be asymptotically stable. The desired field estimate is then obtained by combining the smoothed estimates resulting from each of the one-dimensional problems weighted in an appropriate fashion.