Multiresolution stochastic models and multiscale estimation algorithms

Summary form only given. It has been shown that wavelet transforms and multiscale representations lead naturally to the study of stochastic processes indexed by nodes on lattices and trees, where different depths in the tree or lattice correspond to different spatial scales or resolutions in representing the signal. This framework has been used to develop a theory of modeling for multiscale stochastic processes that leads to a highly nontrivial generalization of Levinson´s algorithm involving recursive generation of models of increasing order, in which the direction of recursion is from coarse to fine resolutions. A theory of optimal estimation for multiresolution stochastic models has been developed. These models lead naturally to several algorithmic structures, one reminiscent of the Laplacian pyramid, one that can be viewed as a multigrid relaxation algorithm, and one that is a generalization of the Rauch-Tung-Striebel algorithm for optimal smoothing of state space models