Resnikoff H.L., Wells R.O. Wavelet analysis and scalable structure of information

New York: Springer, 1998. - 547 p.The authors have been beguiled and entranced by mathematics all of their lives, and both believe it is the highest expression of pure thought and an essential component-one might say the quintessence-of nature. How else can one ex plain the remarkable effectiveness of mathematics in describing and predicting the physical world? The projection of the mathematical method onto the subspace of human endeav 1 ors has long been a source of societal progress and commercial technology. The invention of the electronic digital computer (not the mechanical digital computer of Babbage) has made the role of mathematics in civilization even more central by making mathematics active in the operation of products. The writing of this book was intertwined with the development of a start-up company, Aware, Inc. Aware was founded in 1987 by one of the authors (H.L.R.), and the second author (R.O.W.) put his shoulder to the wheel as a consultant soon after.The Scalable Structure of Information.
The New Mathematical Engineering.
Intriduction.
Trial and Error in the Twenty-first Century.
Active Mathematics.
The Tree Types of Bandwidth.
Good Approximations.
Approximation and the Perception of Reality.
Information Gained from Measurement.
Functions and Their Representation.
Wavelets: A Positional Notation for Functions.
Multiresolution Representation.
The Democratization of Arithmetic.
Music Notation as a Metaphor.
Wavelet Phase Space.
Wavelet Theory.
Algebra and Geometry of Wavelet Matrices.
Introduction.
Wavelet Matrices.
Haar Wavelet Matrices.
Algebraic and Geometric Structure.
Wavelet Matrix Series and Discrete Orthonormal Expansions.
One-Dimensional Wavelet Systems.
Introduction.
The Scaling Equation.
Wavelet Systems.
Recent Developments: Multiwavelets and Lifting.
Examples of One-Dimensional Wavelet Systems.
Introduction to the Examples.
Universal Scaling Functions.
Orthonormal Wavelet Systems.
Flat Wavelets.
Polynomial-regular and Smooth Wavelets.
Fourier Polynomial Wavelet Matrices.
Higher-Dimensional Wavelet Systems.
Introduction.
Scaling Functions.
Scaling Tiles.
Orthonormal Wavelet Bases.
Wavelet Approximation and Algorithms.
The Mallat Algorithm.
Introduction.
Wavelet Series and the Mallat Algorithm.
The Mallat Algorithm for Periodic Data.
Wavelet Approximation.
Introduction.
Vanishing Moments of Wavelet Bases.
Sampling, Reconstruction and Approximation.
Newton's Method.
Biorthogonal Coifman Wavelet Systems.
Comparison with Daubechies Wavelet Systems.
Wavelet Calculus and Connection Coefficients.
An Introduction to Connection Coefficients.
Fundamental Properties of Connection Coefficients.
Wavelet Differentiation.
Algorithms for Computing Connection Coefficients.
Multiscale Representation of Geometry.
Introduction.
Differential Forms and Distributions.
A Multiresolution Representation of Boundary Integration.
Elements of Geometric Measure Theory.
The Wavelet Representations of Integration.
Wavelet-Garlekin Solutions.
Introduction.
Estimates for Differential Equations.
The Dirichlet Problem.
The Neumann Problem for Elliptic Operators.
Iterative Multiscale Methods.
A Wavelet-based Multigrid Iterative Method.
4 Wavelet Applications.
Wavelet Data Compression.
Understanding Compression.
Image Compression.
Transform Image Compression Systems.
Wavelet Image Compression.
Embedded Coding and the Differential-Reduction Wavelet Compression Algorithm.
Multiresolution Audio Compression.
Modulation and Channel Coding.
Understanding Channel Coding.
Multicarrier Communications Systems.
Wavelet Filter Design.
Waveform Design.
Wavelet Channel Coding.
The Wavelet Channel Coding Algorithm.
Wavelet Channel Coding.
Performance of Wavelet Channel Coding.
The DWMT Modem.
Applications and Test Results.