Multiscale Riccati Equations and a Two-Sweep Algorithm for the Optimal Fusion of Multiresolution Data

Abstract

Abstract : In a previous paper [7] we introduced a class of multiscale dynamic models evolving on dyadic trees in which each level in the tree corresponds to the representation of a signal at a particular scale. One of the estimation algorithms suggested in [7] led to the introduction of a new class of Riccati equations describing the evolution of the estimation error covariance as multiresolution data is fused in a fine-to-coarse direction. This equation can be thought of as having 3 steps in its recursive description: a measurement update step, a fine-to-coarse prediction step, and a fusion step. In this paper we analyze this class of equations. In particular by introducing several rudimentary elements of a system theory for processes on trees we develop bounds on the error covariance and use these in analyzing stability and steady-state behavior of the fine-to coarse filter and the Riccati equations. While this analysis is similar in spirit to that for standard Riccati equations and Kalman filters, there are substantial differences that arise in the multiscale context. For example, the asymmetry of the dyadic tree makes it necessary to define multiscale processes via a coarse- to-fine dynamic model and also to define the first step in a fusion processor in the opposite direction - i.e. fine-to-coarse. Also, the notions of stability, reachability, and observability are different. Most importantly for the analysis here, we will see that the fusion step in the fine-to-coarse filter and Riccati equation requires that we focus attention on the maximum likelihood estimator in order to develop a stability and steady-state theory.