Multiscale Recursive Estimation, Data Fusion, and Regularization

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.