Multiscale autoregressive processes. I. Schur-Levinson parametrizations

In many applications 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. A corresponding statistical framework to support the development of optimal, multiscale statistical signal processing algorithms is described. The theory of multiscale signal representation leads naturally to models of signals on trees, and this provides the framework for investigation. In particular, the class of isotropic processes on homogeneous trees is described, and a theory of autoregressive models is developed 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