Generalized Levinson and Schur algorithms for multi-dimensional random field estimation problems

Abstract

Summary form only given. Fast algorithms for computing the linear least-squares estimate of a multidimensional random field from noisy observations inside a circle (2-D) or sphere (3-D) have been derived. The double Radon transform of the random field covariance is assumed to have to Toeplitz-plus-Hankel structure. The algorithms can be viewed as general split Levinson and Schur algorithms, since they exploit this structure in the same way that their one-dimensional counterparts exploit the Toeplitz structure of the covariance of a stationary random process. The algorithm are easily parallelizable, and they are recursive in increasing radius of the hypersphere of observations. A discrete form of the problem and a discrete algorithm for solving it was included. Numerical results on the performance of the algorithm have been obtained. A procedure for estimating a covariance of the desired form from a sample function of a random field (i.e. a multidimensional 'Toeplitzation plus Hankelization') and a one-dimensional discrete algorithm for arbitrary Toeplitz-plus-Hankel systems of equations. >