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Wavelets and their Applications

Book Description

The last 15 years have seen an explosion of interest in wavelets with applications in fields such as image compression, turbulence, human vision, radar and earthquake prediction.

Wavelets represent an area that combines signal in image processing, mathematics, physics and electrical engineering.

As such, this title is intended for the wide audience that is interested in mastering the basic techniques in this subject area, such as decomposition and compression.

Table of Contents

  1. Coverpage
  2. Titlepage
  3. Copyright
  4. Table of Contents
  5. Notations
  6. Introduction
  7. Chapter 1. A Guided Tour
    1. 1.1. Introduction
    2. 1.2. Wavelets
      1. 1.2.1. General aspects
      2. 1.2.2. A wavelet
      3. 1.2.3. Organization of wavelets
      4. 1.2.4. The wavelet tree for a signal
    3. 1.3. An electrical consumption signal analyzed by wavelets
    4. 1.4. Denoising by wavelets: before and afterwards
    5. 1.5. A Doppler signal analyzed by wavelets
    6. 1.6. A Doppler signal denoised by wavelets
    7. 1.7. An electrical signal denoised by wavelets
    8. 1.8. An image decomposed by wavelets
      1. 1.8.1. Decomposition in tree form
      2. 1.8.2. Decomposition in compact form
    9. 1.9. An image compressed by wavelets
    10. 1.10. A signal compressed by wavelets
    11. 1.11. A fingerprint compressed using wavelet packets
  8. Chapter 2. Mathematical Framework
    1. 2.1. Introduction
    2. 2.2. From the Fourier transform to the Gabor transform
      1. 2.2.1. Continuous Fourier transform
      2. 2.2.2. The Gabor transform
    3. 2.3. The continuous transform in wavelets
    4. 2.4. Orthonormal wavelet bases
      1. 2.4.1. From continuous to discrete transform
      2. 2.4.2. Multi-resolution analysis and orthonormal wavelet bases
      3. 2.4.3. The scaling function and the wavelet
    5. 2.5. Wavelet packets
      1. 2.5.1. Construction of wavelet packets
      2. 2.5.2. Atoms of wavelet packets
      3. 2.5.3. Organization of wavelet packets
    6. 2.6. Biorthogonal wavelet bases
      1. 2.6.1. Orthogonality and biorthogonality
      2. 2.6.2. The duality raises several questions
      3. 2.6.3. Properties of biorthogonal wavelets
      4. 2.6.4. Semi-orthogonal wavelets
  9. Chapter 3. From Wavelet Bases to the Fast Algorithm
    1. 3.1. Introduction
    2. 3.2. From orthonormal bases to the Mallat algorithm
    3. 3.3. Four filters
    4. 3.4. Efficient calculation of the coefficients
    5. 3.5. Justification: projections and twin scales
      1. 3.5.1. The decomposition phase
      2. 3.5.2. The reconstruction phase
      3. 3.5.3. Decompositions and reconstructions of a higher order
    6. 3.6. Implementation of the algorithm
      1. 3.6.1. Initialization of the algorithm
      2. 3.6.2. Calculation on finite sequences
      3. 3.6.3. Extra coefficients
    7. 3.7. Complexity of the algorithm
    8. 3.8. From 1D to 2D
    9. 3.9. Translation invariant transform
      1. 3.9.1. ε -decimated DWT
      2. 3.9.2. Calculation of the SWT
      3. 3.9.3. Inverse SWT
  10. Chapter 4. Wavelet Families
    1. 4.1. Introduction
    2. 4.2. What could we want from a wavelet?
    3. 4.3. Synoptic table of the common families
    4. 4.4. Some well known families
      1. 4.4.1. Orthogonal wavelets with compact support
      2. 4.4.1.1. Daubechies wavelets: dbN
        1. 4.4.1.2. Symlets: symN
        2. 4.4.1.3. Coiflets: coifN
      3. 4.4.2. Biorthogonal wavelets with compact support: bior
      4. 4.4.3. Orthogonal wavelets with non-compact support
        1. 4.4.3.1. The Meyer wavelet: meyr
        2. 4.4.3.2. An approximation of the Meyer wavelet: dmey
        3. 4.4.3.3. Battle and Lemarié wavelets: btlm
      5. 4.4.4. Real wavelets without filters
        1. 4.4.4.1. The Mexican hat: mexh
        2. 4.4.4.2. The Morlet wavelet: morl
        3. 4.4.4.3. Gaussian wavelets: gausN
      6. 4.4.5. Complex wavelets without filters
        1. 4.4.5.1. Complex Gaussian wavelets: cgau
        2. 4.4.5.2. Complex Morlet wavelets: cmorl
        3. 4.4.5.3. Complex frequency B-spline wavelets: fbsp
    5. 4.5. Cascade algorithm
      1. 4.5.1. The algorithm and its justification
      2. 4.5.2. An application
      3. 4.5.3. Quality of the approximation
  11. Chapter 5. Finding and Designing a Wavelet
    1. 5.1. Introduction
    2. 5.2. Construction of wavelets for continuous analysis
      1. 5.2.1. Construction of a new wavelet
        1. 5.2.1.1. The admissibility condition
        2. 5.2.1.2. Simple examples of admissible wavelets
        3. 5.2.1.3. Construction of wavelets approaching a pattern
      2. 5.2.2. Application to pattern detection
    3. 5.3. Construction of wavelets for discrete analysis
      1. 5.3.1. Filter banks
        1. 5.3.1.1. From the Mallat algorithm to filter banks
        2. 5.3.1.2. The perfect reconstruction condition
        3. 5.3.1.3. Construction of perfect reconstruction filter banks
        4. 5.3.1.4. Examples of perfect reconstruction filter banks
      2. 5.3.2. Lifting
        1. 5.3.2.1. The lifting method
        2. 5.3.2.2. Lifting and the polyphase method
      3. 5.3.3. Lifting and biorthogonal wavelets
      4. 5.3.4. Construction examples
        1. 5.3.4.1. Illustrations of lifting
        2. 5.3.4.2. Construction of wavelets with more vanishing moments
        3. 5.3.4.3. Approximation of a form by lifting
  12. Chapter 6. A Short 1D Illustrated Handbook
    1. 6.1. Introduction
    2. 6.2. Discrete 1D illustrated handbook
      1. 6.2.1. The analyzed signals
      2. 6.2.2. Processing carried out
      3. 6.2.3. Commented examples
        1. 6.2.3.1. A sum of sines
        2. 6.2.3.2. A frequency breakdown
        3. 6.2.3.3. White noise
        4. 6.2.3.4. Colored noise
        5. 6.2.3.5. A breakdown
        6. 6.2.3.6. Two breakdowns of the derivative
        7. 6.2.3.7. A breakdown of the second derivative
        8. 6.2.3.8. A superposition of signals
        9. 6.2.3.9. A ramp with colored noise
        10. 6.2.3.10. A first real signal
        11. 6.2.3.11. A second real signal
    3. 6.3. The contribution of analysis by wavelet packets
      1. 6.3.1. Example 1: linear and quadratic chirp
      2. 6.3.2. Example 2: a sine
      3. 6.3.3. Example 3: a composite signal
    4. 6.4. “Continuous” 1D illustrated handbook
      1. 6.4.1. Time resolution
        1. 6.4.1.1. Locating a discontinuity in the signal
        2. 6.4.1.2. Locating a discontinuity in the derivative of the signal
      2. 6.4.2. Regularity analysis
        1. 6.4.2.1. Locating a Hölderian singularity
        2. 6.4.2.2. Analysis of the Hölderian regularity of a singularity
        3. 6.4.2.3. Study of two Hölderian singularities
      3. 6.4.3. Analysis of a self-similar signal
  13. Chapter 7. Signal Denoising and Compression
    1. 7.1. Introduction
    2. 7.2. Principle of denoising by wavelets
      1. 7.2.1. The model
      2. 7.2.2. Denoising: before and after
      3. 7.2.3. The algorithm
      4. 7.2.4. Why does it work?
    3. 7.3. Wavelets and statistics
      1. 7.3.1. Kernel estimators and estimators by orthogonal projection
      2. 7.3.2. Estimators by wavelets
    4. 7.4. Denoising methods
      1. 7.4.1. A first estimator
      2. 7.4.2. From coefficient selection to thresholding coefficients
      3. 7.4.3. Universal thresholding
      4. 7.4.4. Estimating the noise standard deviation
      5. 7.4.5. Minimax risk
      6. 7.4.6. Further information on thresholding rules
    5. 7.5. Example of denoising with stationary noise
    6. 7.6. Example of denoising with non-stationary noise
      1. 7.6.1. The model with ruptures of variance
      2. 7.6.2. Thresholding adapted to the noise level change-points
    7. 7.7. Example of denoising of a real signal
      1. 7.7.1. Noise unknown but “homogenous” in variance by level
      2. 7.7.2. Noise unknown and “non-homogenous” in variance by level
    8. 7.8. Contribution of the translation invariant transform
    9. 7.9. Density and regression estimation
      1. 7.9.1. Density estimation
      2. 7.9.2. Regression estimation
    10. 7.10. Principle of compression by wavelets
      1. 7.10.1. The problem
      2. 7.10.2. The basic algorithm
      3. 7.10.3. Why does it work?
    11. 7.11. Compression methods
      1. 7.11.1. Thresholding of the coefficients
      2. 7.11.2. Selection of coefficients
    12. 7.12. Examples of compression
      1. 7.12.1. Global thresholding
      2. 7.12.2. A comparison of the two compression strategies
    13. 7.13. Denoising and compression by wavelet packets
    14. 7.14. Bibliographical comments
  14. Chapter 8. Image Processing with Wavelets
    1. 8.1. Introduction
    2. 8.2. Wavelets for the image
      1. 8.2.1. 2D wavelet decomposition
      2. 8.2.2. Approximation and detail coefficients
        1. 8.2.2.1. Horizontal, vertical and diagonal details
        2. 8.2.2.2. Two representations of decomposition
      3. 8.2.3. Approximations and details
    3. 8.3. Edge detection and textures
      1. 8.3.1. A simple geometric example
      2. 8.3.2. Two real life examples
    4. 8.4. Fusion of images
      1. 8.4.1. The problem through a simple example
      2. 8.4.2. Fusion of fuzzy images
      3. 8.4.3. Mixing of images
    5. 8.5. Denoising of images
      1. 8.5.1. An artificially noisy image
      2. 8.5.2. A real image
    6. 8.6. Image compression
      1. 8.6.1. Principles of compression
      2. 8.6.2. Compression and wavelets
        1. 8.6.2.1. Why does it work?
        2. 8.6.2.2. Why threshold?
        3. 8.6.2.3. Examples of image compression
      3. 8.6.3. “True” compression
        1. 8.6.3.1. The quantization
        2. 8.6.3.2. EZW coding
        3. 8.6.3.3. Comments on the JPEG 2000 standard
  15. Chapter 9. An Overview of Applications
    1. 9.1. Introduction
      1. 9.1.1. Why does it work?
      2. 9.1.2. A classification of the applications
      3. 9.1.3. Two problems in which the wavelets are competitive
      4. 9.1.4. Presentation of applications
    2. 9.2. Wind gusts
    3. 9.3. Detection of seismic jolts
    4. 9.4. Bathymetric study of the marine floor
    5. 9.5. Turbulence analysis
    6. 9.6. Electrocardiogram (ECG): coding and moment of the maximum
    7. 9.7. Eating behavior
    8. 9.8. Fractional wavelets and fMRI
    9. 9.9. Wavelets and biomedical sciences
      1. 9.9.1. Analysis of 1D biomedical signals
        1. 9.9.1.1. Bioacoustic signals
        2. 9.9.1.2. Electrocardiogram (ECG)
        3. 9.9.1.3. Electroencephalogram (EEG)
      2. 9.9.2. 2D biomedical signal analysis
        1. 9.9.2.1. Nuclear magnetic resonance (NMR)
        2. 9.9.2.2. fMRI and functional imagery
    10. 9.10. Statistical process control
    11. 9.11. Online compression of industrial information
    12. 9.12. Transitories in underwater signals
    13. 9.13. Some applications at random
      1. 9.13.1. Video coding
      2. 9.13.2. Computer-assisted tomography
      3. 9.13.3. Producing and analyzing irregular signals or images
      4. 9.13.4. Forecasting
      5. 9.13.5. Interpolation by kriging
  16. Appendix. The EZW Algorithm
    1. A.1. Coding
      1. A.1.1. Detailed description of the EZW algorithm (coding phase)
      2. A.1.2. Example of application of the EZW algorithm (coding phase)
    2. A.2. Decoding
      1. A.2.1. Detailed description of the EZW algorithm (decoding phase)
      2. A.2.2. Example of application of the EZW algorithm (decoding phase)
      3. A.3. Visualization on a real image of the algorithm’s decoding phase
  17. Bibliography
  18. Index