Analogues of Split Levinson, Schur, and Lattice Algorithms for Three-Dimensional Random Field Estimation Problems
A.E. Yagle · SIAM Journal on Applied Mathematics · 1990
Fast algorithms for computing the linear least-squares estimate of a three-dimensional random field from noisy observations inside a sphere are proposed. The algorithms can be viewed as three-dimensional analogues of the split Levinson, Schur, and lattice algorithms of linear prediction, since they exploit an (assumed) Toeplitz-plus-Hankel structure of the double Radon transform of the random field covariance. Therefore these algorithms require fewer computations than would the solution of the three-dimensional Wiener-Hopf integral equation. Unlike previous generalized Levinson algorithms, no quarter-plane or asymmetric half-plane support assumptions for the filter are necessary; nor is the three-dimensional filtering problem treated as a multichannel (vector) filtering problem. The algorithms work in three stages. First, the three-dimensional split Schur algorithm computes a potential from the covariance of the random field. This potential is a three-dimensional analogue of the parameter appearing in the split Levinson algorithm. Alternatively, the three-dimensional split lattice algorithm may be used to compute the potential from the canonical spectral factor of the covariance of the observation field. Next, the three-dimensional split Levinson algorithm computes the Radon transform of the three-dimensional prediction filter for estimating the random field on the surface of the sphere of noisy observations. Finally, this filter is used to compute the smoothing filter for estimating the random field inside the sphere of observations. The algorithms generalize known results for isotropic, two-dimensional random fields.