MULTISCALE AUTOREGRESSIVE PROCESSES .1. SCHUR-LEVINSON PARAMETRIZATIONS

In many applications (e.g., recognition of geophysical and biomedical signals and multiscale analysis of images), it is of interest to analyze and recognize phenomena occurring at different scales. The recently introduced wavelet transforms provide a time-and-scale decomposition of signals that offers the possibility of such analysis. At present, however, there is no corresponding statistical framework to support the development of optimal, multiscale statistical signal processing algorithms. In this paper we describe such a framework. The theory of multiscale signal representations leads naturally to models of signals on trees, and this provides the framework for our investigation. In particular, in this paper we describe the class of isotropic processes on homogeneous trees and develop a theory of autoregressive models in this context. This leads to generalizations of Schur and Levinson recursions, associated properties of the resulting reflection coefficients, and the initial pieces in a system theory for multiscale modeling.