Wavelet thresholding and noise reduction

Maarten Jansen

Table of contents

• Abstract

• Voorwoord - Preface

• Nederlandse Samenvatting

• Contents

• Notations and Abbreviations

• List of Figures

• List of Tables

• Chapter 1: Introduction and overview

  • Notions and notations
    • Mathematical preliminaries
    • Fourier analysis and digital signals
    • A note on images
  • Outline of this thesis
  • Motivation
  
  

• Chapter 2: Wavelets and wavelet thresholding

  • Exploiting sample correlations
    • The input problem: sparsity
    • Basis functions and multiresolution
    • The dilation equation
    • (Fast) Wavelet Transforms and Filter Banks
    • Locality
    • Vanishing moments
    • Two-dimensional wavelet transforms
  • Continuous wavelet transform
  • Non-decimated wavelet transforms and frames
  • Lifting and second generation wavelets
    • The idea behind lifting
    • The integer wavelet transform
    • Non-equidistant data
  • Noise reduction by thresholding wavelet coefficients
    • Noise model and definitions
    • The wavelet transform of a signal with noise
    • Wavelet thresholding, motivation
    • Hard- and soft-thresholding, shrinking
    • Threshold assessment
    • Thresholding as non-linear smoothing
  • Other coefficient selection principles
  • Basis selection methods
  • Wavelets in other domains of application
  • Summary and concluding remarks
  
  

• Chapter 3: The minimum mean squared error threshold

  • Mean square error and Risk function
    • Definitions
    • Variance and bias
  • The risk contribution of each coefficient (Gaussian noise)
  • The asymptotic behavior of the minimum risk threshold for piecewise polynomials
    • Motivation
    • Asymptotic equivalence
    • The asymptotic behavior
    • An example
    • Why does the threshold depend on the number of data points?
  • Universal Threshold
    • Oracle mimicking
    • Minimax properties
    • Adaptivity, optimality within function classes
    • Smoothness
    • Probabilistic Upper bound
    • Universal threshold in practice
  • Beyond the piecewise polynomial case
    • For which coefficients is a given threshold too large/small?
    • Intermediate results for the risk in one coefficient
    • Piecewise smooth functions
    • Function spaces
  • Conclusion
  
  

• Chapter 4: Estimating the minimum MSE threshold

  • SURE, a first estimator for the MSE
    • The effect of the threshold operation
    • Counting the number of coefficients below the threshold
    • SURE is adaptive
  • Ordinary Cross Validation
  • Generalized Cross Validation
    • Definition
    • Asymptotic behavior
      • A quotient of two random variables
      • Limit behavior of the upper bound
      • General piecewise smooth signals
      • Conclusion
  • GCV for a finite number of data
    • The minimization procedure
    • Convexity and continuity
    • Behavior for large thresholds and problems near the origin
    • GCV in absence of signal and in absence of noise
    • Absolute and relative error
    • Which is better: GCV or universal?
  • Concluding remarks
  
  

• Chapter 5: Thresholding and GCV applicability in more realistic situations

  • Scale dependent thresholding
    • Correlated noise
    • Non-orthogonal transforms
    • Scale-adaptivity
  • Tree-structured thresholding
  • Non-decimated wavelet transforms
  • Test examples and comparison of different methods
    • Orthogonal transform, white noise
    • Biorthogonal transform, colored noise
  • Integer wavelet transforms
  
  

• Chapter 6: Bayesian correction with geometrical priors for image noise reduction

  • An approximation theoretic point of view
    • Step function approximation in one dimension
    • Approximations in two dimensions
    • Smoothness spaces
    • Other basis functions
  • The Bayesian approach
    • Motivation and objectives
    • Plugging the threshold procedure into a fully random \\ model
    • Threshold mask images
    • Binary image enhancement methods
    • Bayesian classification
  • Prior and conditional model
    • The prior model
    • The conditional model
  • The Bayesian algorithm
    • Posterior probabilities
    • Stochastic sampling
  • Parameter estimation
    • Parameters of the conditional model
    • Full Bayes or empirical Bayes
  • The algorithm and its results
    • Algorithm overview
    • Results and discussion
    • Related methods
  • Summary and conclusions
  
  

• Chapter 7: Smoothing non-equidistantly spaced data using second generation wavelets and thresholding

  • Thresholding second generation coefficients
    • The model and procedure
    • Threshold selection
    • Two examples
  • The bias
    • The problem
    • Condition of the wavelet transform
    • Where does the bad condition come from?
  • How to deal with the bias?
    • Computing the impact of a threshold
    • Hidden components and correlation between coefficients
    • Starting from a first-generation solution
    • The proposed algorithm
    • Results and discussion
  
  

• Chapter 8: Overview of contribution and concluding remarks

  • Contribution
  • Open problems and suggestions for further research
    • Non-Gaussian noise
    • Bayesian correction
    • Stable transformations for non-equispaced data
  
  

• Bibliography

• Index