Multiresolution Markov models for signal and image processing

This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts-in particular, making ties to topics such as wavelets and multigrid methods. A third goal is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden Markov models. We also discuss the limitations of tree-based models and algorithms and the artifacts that they can introduce. We describe when these are of concern and ways in which they can be overcome. This leads to a discussion of MR models on more general graphs and ties to well-known and emerging methods for inference on graphical models.