Beyond Wavelets 1st Edition by Grant Welland ISBN 1435631994 9781435631991
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ISBN 10: 1435631994
ISBN 13: 9781435631991
Author: Grant Welland
Morlet and Grossmann and developed very rapidly during the 1980’s and 1990’s, has created a common link between computational mathematics and other disciplines of science and engineering.
Classical wavelets have provided effective and efficient mathematical tools for time-frequency analysis which enhances and replaces the Fourier approach.
However, with the current advances in science and technology, there is an immediate need to extend wavelet mathematical tools as well. “Beyond Wavelets” presents a list of ideas and mathematical
foundations for such extensions, including: continuous and digital ridgelets, brushlets, steerable wavelet packets, contourlets, eno-wavelets, spline-wavelet frames, and quasi-affine wavelets. Wavelet subband algorithms are extended to pyramidal directional and nonuniform filter banks. In addition, this volume includes a
method for tomographic reconstruction using a mechanical image model and a statistical study for independent adaptive signal representation.
Investigators already familiar with wavelet methods from areas such as engineering, statistics, and mathematics will benefit by owning this volume.
*Curvelets, Contourlets, Ridgelets,
*Digital Implementation of Ridgelet Packets
*Steerable Wavelet Packets
*Essentially Non-Oscillatory Wavelets
*Medical Imaging
*Non-Uniform Filter Banks
*Spline-wavelet frames and
*Vanishing Moment Recovery Functions
Beyond Wavelets 1st Table of contents:
Chapter 1: Digital Ridgelet Transform Based on True Ridge Functions
Abstract
1.1 Introduction
1.2 Digital Ridgelets
1.3 Relation to Fast Slant Stack
1.4 Structural Analogies
1.5 Example: Halfdome
1.6 Sparsity of the Frame Kernel
1.7 Comparisons
1.8 Discussion
Acknowledgement
Chapter 2: Digital Implementation of Ridgelet Packets
Abstract
2.1 Introduction
2.2 Fourier Preliminaries
2.3 Radon Preliminaries
2.4 The Ridgelet Construction, And Its Properties
2.5 Ridgelet Packet Construction
2.6 Implementation On Digital Data
2.7 Adaptation
2.8 Discussion
Acknowledgement
Chapter 3: Brushlets: Steerable Wavelet Packets
Abstract
3.1 Introduction
3.2 Biorthogonal Windowed Fourier Bases
3.3 Choice of the Bell Function
3.4 Biorthogonal Brushlet Bases
3.5 Conclusion
Chapter 4: Contourlets
Abstract
4.1 Introduction and Motivation
4.2 Representing 2-D Piecewise Smooth Functions
4.3 Pyramidal Directional Filter Bank
4.4 Multiresolution Analysis
4.5 Numerical Experiments
4.6 Conclusion
Acknowledgement
Chapter 5: Eno-Wavelet Transforms and Some Applications
Abstract
5.1 Introduction
5.2 The Eno-Wavelet Algorithm
5.3 Theory: Error Bound And Stability
5.4 Applications
Acknowledgements
Chapter 6: A Mechanical Image Model for Bayesian Tomographic Reconstruction
Abstract
6.1 Introduction and Background
6.2 Materials and Methods
6.3 Results and Discussion
Acknowledgements
Chapter 7: Recent Development of Spline Wavelet Frames with Compact Support
Abstract
7.1 Introduction
7.2 Characterization of Wavelet Spline Frames
7.3 Wavelet Frames of Splines with Multiple Knots
7.4 The Common Link: Approximate Duals
7.5 Tight Spline Frames with Non-Uniform Knots
7.5.1 Piecewise Linear Tight Frames
7.5.2 Piecewise Cubic Tight Frames with Equidistant Simple Knots
7.5.3 Tight Frames of Cubic Splines with Equidistant Knots of Multiplicity 2
Appendix
Acknowledgement
Chapter 8: Affine, Quasi-Affine and Co-Affine Wavelets
Abstract
8.1 Introduction
8.2 Frames and the Three Systems X(ψ), X*(ψ), and
Acknowledgement
Chapter 9: Sparsity vs. Statistical Independence in Adaptive Signal Representations: A Case Study of the Spike Process
Abstract
9.1 Introduction
9.2 Notation And Terminology
9.3 Sparsity vs. Statistical Independence
9.4 Two-Dimensional Counterexample
9.5 The Spike Process
9.6 Proofs Of Propositions And Theorems
9.7 Discussion
Acknowledgement
9.8 Appendices
Chapter 10: Nonuniform Filter Banks: New Results and Open Problems
Abstract
10.1 Introduction
10.2 Background: Equivalent Uniform FBs; PR Equations
10.3 Problem Statement, and Unconstrained FBs
10.4 Tree Structures
10.5 Delay-Chains
10.6 The Class of Rational FBs
10.7 Conditions Based on Reductions to Tree Structures
10.8 Summary And Comparison Of Necessary Conditions
10.9 Concluding Remarks
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