Beyond Wavelets (Volume 10) (Studies in Computational Mathematics, Volume 10)

✍ Scribed by Grant Welland (editor)

• Publisher: Academic Press
• Year: 2003
• Tongue: English
• Leaves: 321
• Edition: 1
• Category: Library

✦ Synopsis

"Beyond Wavelets" presents state-of-the-art theories, methods, algorithms, and applications of mathematical extensions for classical wavelet analysis. Wavelets, introduced 20 years ago by
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 PacketsSteerable Wavelet PacketsEssentially Non-Oscillatory WaveletsMedical ImagingNon-Uniform Filter BanksSpline-wavelet frames and Vanishing Moment Recovery Functions

✦ Table of Contents

Front Cover
Beyond Wavelets
Copyright Page
Contents
Preface
Chapter 1. Digital Ridgelet Transform based on True Ridge Functions
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
References
Chapter 2. Digital Implementation of Ridgelet Packets
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
References
Chapter 3. Brushlets : Steerable Wavelet Packets
3.1 Introduction
3.2 Biorthogonal windowed Fourier bases
3.3 Choice of the bell function
3.4 Biorthogonal brushlet bases
3.5 Conclusion
References
Chapter 4. Contourlets
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
References
Chapter 5. ENO-wavelet Transforms and Some Applications
5.1 Introduction
5.2 The ENO-Wavelet Algorithm
5.3 Theory: Error Bound and Stability
5.4 Application
References
Chapter 6. A Mechanical Image Model for Bayesian Tomographic Reconstruction
6.1 Introduction and Background
6.2 Materials and Methods
6.3 Results and Discussion
References
Chapter 7. Recent Development of Spline Wavelet Frames with Compact Support
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
References
Chapter 8. Afiine, Quasi-Affine and Co-Affine Wavelets
8.1 Introduction

References
Chapter 9. Sparsity vs. Statistical Independence in Adaptive Signal Representations: A Case Study of the Spike Process
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
9.8 Appendices
References
Chapter 10. Nonuniform Filter Banks: New Results and Open Problems
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
10.10 Appendices
References
Index