MULTISCALE RECURSIVE ESTIMATION, DATA FUSION, AND REGULARIZATION

A current topic of great interest is the multiresolution analysis of s ignals and the development of multiscale signal processing algorithms. In this paper, we describe a framework for modeling stochastic phenom ena at multiple scales and for their efficient estimation or reconstru ction given partial and/or noisy measurements which may also be at sev eral scales. In particular multiscale signal representations lead natu rally 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 clas s of multiscale dynamic models evolving on dyadic trees. The main focu s of this paper is on the description, analysis, and application of an extremely efficient optimal estimation algorithm for this class of mo dels. This algorithm consists of a fine-to-coarse filtering sweep, fol lowed by a coarse-to-fine smoothing step, corresponding to the dyadic tree generalization of Kalman filtering and Rauch-Tung-Striebel smooth ing. 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 t ime-(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 si gnal and image processing problems encountered, for example, in low-le vel computer vision.