Identifying multiscale statistical models using the wavelet transform

2 IDENTIFYING MULTISCALE STATISTICAL MODELS USING THE WAVELET TRANSFORM I by Stuart A. Golden Submitted to the Department of Electrical Engineering on Tuesday, April 23, 1991 in partial fulfillment of the requirements for the Degrees of Electrical Engineer and Master of Science in Electrical Engineering ABSTRACT Recently much attention has been focused on methods for performing multiple resolution decompositions of signals based on wavelet transforms. In this thesis we develop an algorithm to determine optimal wavelet transforms based upon the statistical characterization of the signal being analyzed. Our criterion is based upon the desire to find the optimal wavelet transform approximation of a KarhunenLoeve expansion, i.e. we would like the transformed coefficients to be as close to white as possible. We determine the optimal wavelet transform in a level-by-level procedure. Using Vaidyanathan and Hoang's parameterization [11 of quadrature mirror filters (QMFs), we chose the QMF pair such that the I-level coarser approximation of the signal and the wavelet coefficients at that level are as close to being statistically uncorrelated with each other as possible. This procedure is then repeated for an arbitrary number of levels. We examine the ability of the transform to achieve an approximate Karhunen-Loeve expansion by considering several examples. The examples that we consider are a first-order Gauss-Markov process, a second-order under-damped process, and fractional Brownian motions. Thesis Supervisor: Dr. Alan S. Willsky Title: Professor of Electrical Engineering Acknowiedzernents 3 Acknowledgements I would like to thank my advisor Professor Alan Willsky for his continual guidance and support throughout this endeavor. I feel that a large portion of my learning at MIT is a direct result of working with Prof. Willsky. I am also very grateful for the support and assistance of my fellow graduate students. In particular, I greatly appreciate the times that Ken Chou and Darrin Taylor have spent in helping me answer some of my many questions.

[1]  K. B. Haley,et al.  Optimization Theory with Applications , 1970 .

[2]  P. P. Vaidyanathan,et al.  Lattice structures for optimal design and robust implementation of two-channel perfect-reconstruction QMF banks , 1988, IEEE Trans. Acoust. Speech Signal Process..

[3]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[4]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[5]  A.H. Tewfik,et al.  Correlation structure of the discrete wavelet coefficients of fractional Brownian motion , 1992, IEEE Trans. Inf. Theory.

[6]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[7]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[8]  C. Striebel,et al.  On the maximum likelihood estimates for linear dynamic systems , 1965 .

[9]  Eero P. Simoncelli.  Orthogonal sub-band image transforms , 1988 .

[10]  Mark J. T. Smith,et al.  Exact reconstruction techniques for tree-structured subband coders , 1986, IEEE Trans. Acoust. Speech Signal Process..

[11]  Alan S. Willsky,et al.  Modeling and estimation of multiscale stochastic processes , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[12]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[13]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[14]  Gregory W. Wornell,et al.  Estimation of fractal signals from noisy measurements using wavelets , 1992, IEEE Trans. Signal Process..

[15]  Alex Pentland,et al.  Fractal-Based Description of Natural Scenes , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  Gregory W. Wornell,et al.  A Karhunen-Loève-like expansion for 1/f processes via wavelets , 1990, IEEE Trans. Inf. Theory.

[17]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[18]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[19]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[20]  P. P. Vaidyanathan,et al.  Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial , 1990, Proc. IEEE.

[21]  D. J. Paddon,et al.  Multigrid Methods for Integral and Differential Equations. , 1987 .

[22]  Stéphane Mallat,et al.  Multifrequency channel decompositions of images and wavelet models , 1989, IEEE Trans. Acoust. Speech Signal Process..

[23]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[24]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .