Modeling and Estimation of Multiresolution Stochastic Processes
Abstract
In this paper, we provide an overview of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As we describe, a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular we will see how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of "multiscale stationarity" in which scale plays the role of a time-like variable. In this paper we focus primarily on the investigation of models on homogeneous trees. In particular we describe the elements of a dynamic system theory on trees and introduce two notions of stationarity. One of these leads naturally to the development of a theory of multiscale autoregressive modeling including a generalization of the celebrated Schur and Levinson algorithms for order-recursive model building. The second, weaker motion of stationarity leads directly to a class of state space models on homogeneous trees. We describe several of the elements of the system theory for such models and also describe the natural, extremely efficient algorithmic structures for optimal estimation that these models suggest: one class of algorithms has a multigrid relaxation structure; a second uses the scale-to-scale whitening property of wavelet transforms for our models; and a third leads to a new class of Riccati equations involving the usual predict and update steps and a new "fusion" step as information is propagated from fine to coarse scales. As we will see, this framework allows us to consider in a very natural way the fusion of data from sensors with differing resolution
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1991
- Accession Number
- ADA459289
Entities
People
- Alan S. Willsky
- Albert Benveniste
- Kenneth C. Chou
- Michele Basseville
- Ramine Nikoukhah
- Stuart A. Golden
Organizations
- Massachusetts Institute of Technology