Fast algorithms for linear least squares smoothing problems in one and two dimensions using generalized discrete Bellman-Siegert-Krein resolvent identities

New fast algorithms for linear least squares smoothing problems in one and two dimensions are derived. These are discrete Manuscript received November 9, 1989; revised March 23, 1991. This work was supported by the Air Force Office of Scientific Research under grant AFOSR-89-0017. The authors are with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122. IEEE Log Number 9107643. and multidimensional generalizations of the Bellman-Siegert-Krein resolvent identity, which has been applied to the continuous, one-dimensional stationary smoothing problem by Kailath. The new equations relate the linear least squares prediction filters associated with discrete random fields to the smoothing filters for those fields. This results in new fast algorithms for deriving the latter from the former. In particular, used in conjunction with recently developed generalized one(two-) dimensional split Levinson and Schur algorithms for covariances with (block) Toeplitz-plus-Hankel structure, these algorithms can be used to compute smoothing filters for random fields defined on a polar raster, using fewer computations than those required by previous algorithms.