Abstruet-A current topic of great interest is the multiresolution analysis of signals and the development of multiscale signal processing algorithms. In this paper, we describe a framework for modeling stochastic phenomena at multiple scales and for their efficient estimation or reconstruction given partial and/or noisy measurements which may also be at several scales. In particular multiscale signal representations lead naturally to pyramidal or tree-like data structures in which each level in the tree corresponds to a particular scale of representation. Noting that scale plays the role of a time-like variable, we introduce a class of multiscale dynamic models evolving on dyadic trees. The main focus of this paper is on the description, analysis, and application of an extremely efficient optimal estimation algorithm for this class of models. This algorithm consists of a fine-tocoarse filtering sweep, followed by a coarse-to-fine smoothing step, corresponding to the dyadic tree generalization of Kalman filtering and Rauch-Tung4triebel smoothing. The Kalman filtering sweep consists of the recursive application of three steps: a measurement update step, a fine-to-coarse prediction step, and a fusion step, the latter of which has no counterpart for time-(rather than scale-) recursive Kalman filtering. We illustrate the use of our methodology for the fusion of multiresolution data and for the efficient solution of "fractal regularizations" of ill-posed signal and image processing problems encountered, for example, in low-level computer vision.

Abstract-An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As described, a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it will be seen how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of " multiscale stationarity" in which scale plays the role of a time-like variable. Focus is primarily on the investigation of models on homogenous trees. In particular, the elements of a dynamic system theory on trees are described and two notions of stationarity are introduced. One of these leads naturally to the development of a theory of multiscale autoregressive modeling including a generalization of the celebrated Schur and Levinson algorithms for order-recursive model building. The second, weaker notion of stationarity leads directly to a class of state space models on homogenous trees. Several of the elements of the system theory for such models are described and also the natural, extremely efficient algorithmic structures for optimal estimation are described that these models suggest: one class of algorithms has a multigrid relaxation structure; a second uses the scale-to-scale whitening property of wavelet transforms for our models; and a third leads to a new class of Riccati equations involving the usual predict and update steps and a new "fusion" step as information is propagated from fine to coarse scales. This framework allows for consideration, in a very natural way, the fusion of data from sensors with differing resolutions. Also, thanks to the fact that wavelet transforms do an excellent job of "compressing" large classes of covariance kernels, it will be seen that these modeling paradigms appear to have promise in a far broader context than one might expect.

A current topic of great interest is the multi-resolution analysis of signals and the development of multi-scale or multigrid algorithms. In this paper we describe part of a research effort aimed at developing a corresponding theory for stochastic processes described at multiple scales and for their efficient estimation or reconstruction given partial and/or noisy measurements which may also be at several scales. The theories of multi-scale signal representations and wavelet transforms lead naturally to models of signals(in one or several dimensions) on trees and lattices. In this paper we focus on one particular class of processes defined on dyadic trees. The central results of the paper are three algorithms for optimal estimation/reconstruction for such processes: one reminiscent of the Laplacian pyramid and the efficient use of Haar transforms, a second that is iterative in nature and can be viewed as a multigrid relaxation algorithm, and a third that represents an extension of the Rauch-Tung-Striebel algorithm to processes on dyadic trees and involves a new discrete Riccati equation, which in this case has three steps: predict, merge, and measurement update.Related work and extensions are also briefly discussed.