An overlapping tree approach to multiscale stochastic modeling and estimation

TitleAn overlapping tree approach to multiscale stochastic modeling and estimation
Publication TypeJournal Article
Year of Publication1997
AuthorsIrving, W. W., P. Fieguth, and A. S. Willsky
JournalIEEE Transactions on Image Processing
Pagination1517 - 1529
Keywordscovariance analysis, covariance structure, error analysis, estimation algorithm, estimation error covariances, fine scale detail resolution, image domain, image reconstruction, image resolution, least squares approximations, least squares estimation, multiscale stochastic modeling, overlapping tree, parameter estimation, quadtrees, random field estimation, random processes, scale recursive dynamic trees, second order statistics, statistically optimal algorithm, stochastic processes

Recently, a class of multiscale stochastic models has been introduced in which random processes and fields are described by scale-recursive dynamic trees. A major advantage of this framework is that it leads to an extremely efficient, statistically optimal algorithm for least-squares estimation. In certain applications, however, estimates based on the types of multiscale models previously proposed may not be adequate, as they have tended to exhibit a visually distracting blockiness. We eliminate this blockiness by discarding the standard assumption that distinct nodes on a given level of the multiscale process correspond to disjoint portions of the image domain; instead, we allow a correspondence to overlapping portions of the image domain. We use these so-called overlapping-tree models for both modeling and estimation. In particular, we develop an efficient multiscale algorithm for generating sample paths of a random field whose second-order statistics match a prespecified covariance structure, to any desired degree of fidelity. Furthermore, we demonstrate that under easily satisfied conditions, we can ldquo;lift rdquo; a random field estimation problem to one defined on an overlapped tree, resulting in an estimation algorithm that is computationally efficient, directly produces estimation error covariances, and eliminates blockiness in the reconstructed imagery without any sacrifice in the resolution of fine-scale detail