Wavelets: Software and Applications

Table of Contents

• Preface
• Nederlandse Samenvatting
• Contents
• Notations and Abbreviations
• List of Figures
• List of Tables
• Chapter 1: Introduction
  • Overview of Wavelet Research
  • Overview of the Thesis
  • Situation of the Research
• Chapter 2: Wavelets
  • What are Wavelets?
  • Why wavelets?
  • Wavelets and Multi-Resolution Analysis
    • Multi-Resolution Analysis
    • Wavelet Functions
    • The Fast Wavelet Transform (FWT)
    • Orthogonal Wavelets
    • Biorthogonal Wavelets
    • The Wavelet Decomposition Tree
    • Higher Dimensions
  • The Wavelet Transform and Translation Invariance
  • The Redundant Wavelet Transform
  • Wavelet Packets
• Chapter 3: The Lifting Scheme
  • Introduction
  • Predict and Update
    • The Inverse Transform
    • Example: Linear Prediction
    • Flexibility
  • Unifying the Two Paradigms
    • The Filter Bank Algorithm
    • Polyphase Representations
    • Primal and Dual Lifting
    • Decomposition into Lifting Steps
    • Advantages of the Lifting Scheme
  • The Integer Wavelet Transform
    • Integer Transforms
    • Rational Lifting Coefficients
    • Normalization
  • Boundary Treatment
    • Classical Extension Methods
    • Signal Extension with Lifting
  • Conclusions
• Chapter 4: Wavelet-Based Image Processing
  • Digital Images
    • Discretization
    • Quantization
    • Color images
    • Generalization
    • L^*a^*b^*
  • Wavelet Denoising
    • Thresholding
    • Threshold selection
    • Integer Minimization Problem
  • Image Scaling
  • Crop and Merge
  • Image Compression
    • Why Compression?
    • Compression Techniques
    • Features of Image Compression using Wavelets
    • Zerotree Encoding
  • Conclusions
• Chapter 5: Design of the Software Library WAILI
  • Design Criteria
  • WAILI: Wavelets with Integer Lifting
    • The Lifting Scheme
    • Cohen-Daubechies-Feauveau Biorthogonal Wavelets
    • Image Size Constraints
    • Images and Wavelet Transforms
  • Wavelet-Based Operations
    • Crop and Merge on Wavelet-Transformed Images
    • Scaling of Images
    • Noise Reduction
  • Implementation Issues
    • In-Place Transform
    • The Two-Dimensional Wavelet Transform: Mallat Ordering
    • Transform Order
    • Normalization of the Wavelet Coefficients
  • Summary of the Functionality of WAILI
  • Evaluation of the Compression Potential
    • Description of the Compression Method
    • Measure of the Compression Rate
    • Measure of the Image Quality
    • Results
  • Comparison with Other Wavelet Software
• Chapter 6: Large-Scale Image Processing
  • Introduction
  • Very Large Images and Tiling
    • Tiling
    • Block Management
    • Implementation in WAILI
  • Image Processing on Tiled Images
    • Block-Based Denoising
    • Image Compression
    • Crop and Merge
    • A GIS Application
  • Transform Results
  • Conclusions
• Chapter 7: The Red-Black Wavelet Transform
  • Introduction
  • The Red-Black Wavelet Transform
    • A Two-Step Method
    • Borders
    • Reordering
    • Integer Wavelet Transform
    • Basis Functions
    • Other Members of the Family
  • Properties of the Red-Black Wavelet Transform
  • The Redundant Wavelet Transform
  • Comparison with the CDF (2, 2) Tensor Product Wavelets
    • Test case 1
    • Test case 2
  • Conclusions
• Chapter 8: Experiments with a Wavelet-Based APOD
  • Introduction
  • The Proper Orthogonal Decomposition
  • The Approximate Proper Orthogonal Decomposition
    • Theory of Operation
    • Decorrelating Transforms
    • Decomposition into the Basis of Approximate Eigenmodes
  • Wavelet Transforms
  • Results
    • Wavelets
    • The Dynamical System
    • Snapshots
    • ``Strange'' Snapshots
    • Eigenvalue Distribution
    • Eigenmodes
    • Reconstruction of the Snapshots
    • Coefficients of the Snapshots in the APOD Basis
    • Timings
  • Conclusions
• Chapter 9: Conclusions and Suggestions for Future Research
  • Wavelet Transforms
  • A Software Library for Wavelet-Based Image Processing
  • Approximate POD using Wavelet Packets
  • Suggestions for Future Research
• Bibliography
• Appendix A: Lifting Decompositions
• Appendix B: A Simple Image Compression Example using WAILI
• Index