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 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

in signal and image processing, including image denoising, coding, and super-resolution. # 2001 Academic Press 1. INTRODUCTION Stochastic models of natural images underlie a variety of applications in image processing and low-level computer vision, including image coding, denoising and 1 MW supported by NSERC 1967 fellowship; AW and MW by AFOSR Grant F49620-98-1-0349 and ONR Grant N00014-91-J-1004. Address correspondence to MW. 2 ES supported by NSF Career Grant MIP-9796040 and an Alfred P. Sloan fellowship. 89 1063-5203/01 $35.00 Copyright # 2001 by Academic Press All rights of reproduction in any form reserved. 90 WAINWRIGHT, SIMONCELLI, AND WILLSKY restoration, interpolation and synthesis. Accordingly, the past decade has witnessed an increasing amount of research devoted to developing stochastic models of images (e.g., [19, 38, 45, 48, 55]). Simultaneously, wavel

Abstract. We address the problem of modeling the spatial and temporal second-order statistics of video sequences that exhibit both spatial and temporal regularity, intended in a statistical sense. We model such sequences as dynamic multiscale autoregressive models, and introduce an efficient algorithm to learn the model parameters. We then show how the model can be used to synthesize novel sequences that extend the original ones in both space and time, and illustrate the power, and limitations, of the models we propose with a number of real image sequences. 1

In this paper we develop a stochastic realization theory for multiscale autoregressive (MAR) processes that leads to computationally efficient realization algorithms. The utility of MAR processes has been limited by the fact that the previously known general purpose realization algorithm, based on canonical correlations, leads to model inconsistencies and has complexity quartic in problem size. Our realization theory and algorithms addresses these issues by focusing on the estimation-theoretic concept of predictive efficiency and by exploiting the scale-recursive structure of so-called internal MAR processes. Our realization algorithm has complexity quadratic in problem size and with an approximation we also obtain an algorithm that has complexity linear in problem size.

to my wife and my parents with love Acknowledgments I have a long list of people to thank for the support and guidance. First, I want to thank my parents William N. Slatton and Linda Matros. They inspired me to pursue an education and supported me in many ways during that education. They were by far my most important influences during the first twenty years of my life. I also must thank my wonderful wife Jennifer for giving her love and sup-port. It was not easy to be married to a graduate student. She sacrificed her standard of living and more importantly her time with me while I remained in school. She also brought a glorious new life, our son William H. Slatton, into the world during this time. For all of this, I will always be grateful and in awe of her. I wish to thank my advisors, Prof. Melba M. Crawford and Prof. Brian L. Evans. Without their insightful guidance and words of encouragement this disser-tation would never have been realized. I also wish to thank other people that have served as teacher to me at differ-ent stages in my graduate education. Dr. Yunjin Kim was extremely generous with his time and helped me refine my research topic, which eventually led to my receiv-ing a National Aeronautics and Space Administration Graduate Research Program Fellowship. Dr. Robert Treuhaft was also giving of his time when I struggled to understand certain radar scattering problems.

In data assimilation for a system which evolves in time, one combines past and current observations with a model of the dynamics of the system, in order to improve the simulation of the system as well as any future predictions about it. From a statistical point of view, this process can be regarded as estimating many random variables, which are related both spatially and temporally: given observations of some of these variables, typically corresponding to times past, we require estimates of several others, typically corresponding to future times.

Local "belief propagation" rules of the sort proposed byPearl [12] are guaranteed to converge to the correct posterior probabilities in singly connected graphical models. Recently, a number of researchers have empirically demonstrated good performance of "loopy belief propagation" -- using these same rules on graphs with loops. Perhaps the most dramatic instance is the near Shannonlimit performance of "Turbo codes", whose decoding algorithm is equivalentto loopy belief propagation. Except for the

The Hypertext Transfer Protocol (HTTP) and the File Transfer Protocol (FTP) are both application-level protocols layered over the Transmission Control Protocol (TCP). HTTP is a request/response protocol used for the data transfer over the Internet. The primary function of FTP, on the other hand, is defined as transferring files efficiently and reliably among hosts. We study the statistical and dynamical properties of both HTTP and FTP traffic. We use Network Simulator (NS) to generate synthesized data for HTTP and FTP traffic.

this paper we describe a system to show some limited effects on a static toy-car model and present techniques that can be used in similar setups. Our focus is on creating apparent motion for animation

The posterior sampling problem computes a random sample from a posterior distribution. Typically this problem is solved through Markov-Chain Monte-Carlo / Simulated Annealing, however these can be computationally challenging and slow to converge. In this paper we use a little-known property of multiscale statistical models to formulate a posterior sampler, exact in the case of Markov random fields, and approximate for other distributions. The proposed approach benefits from and builds upon past work on multiscale model inference and approximation, yielding a fast approach to sampling continuous-state images.