Random Sampling of Random Fields: Least Squares Estimation
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The paper begins with a discussion of deterministic sampling, where it is observed that when one can reconstruct the covariance one can also reconstruct the sample path (in quadratic mean). Then the theorem of Shapiro and Silverman, which states that Poisson based sampling allows reconstruction of the covariance at any sampling rate and a construction of an estimator of the covariance (due to Papoulis) are presented. A class of estimators for random fields using Poisson (and Poisson like) sampling is developed. The optimal estimator (minimum mean square error) is shown to exist and the error is shown to go to zero only as the sampling rate goes to infinity; Poisson sampling behaves differently from regular sampling in this respect. Poisson sampling is shown to be the best (lowest error) for a wide class of multidimensional point processes (sampling meassures). One feature of the development is that it applies directly in IR (Euclidian N-Space). It is shown that the optimal estimator has many desirable properties (continuity, etc.); however, recursion in terms of the density of the sampling processes is not easily developed. A sub-optimal estimator with this desirable property is also discussed. In the case that the random field is Gaussian, the proposed estimator is seen to be the conditional mean.