Image Processing and Data Analysis: The Multiscale Approach

Image Processing and Data Analysis: The Multiscale Approach

Author(s) : Jean-Luc Starck, Fionn D. Murtagh, Albert Bijaoui
ISBN : 0521599148
Pages : 297
Publication Date : July 1998
Publisher: Cambridge University Press

Number of techniques have been developed in recent years for the analysis of digital data, especially the manipulation of images. This book provides an in-depth introduction to a range of these data-processing techniques. It develops the reader’s understanding of each technique and then shows with practical examples how they can be applied to improve the skills of graduate students and researchers in astronomy, electrical engineering, physics, geophysics and medical imaging.

What sets this book apart from others on the subject is the complementary blend of theory and practical application. Throughout, the book is illustrated with real-world examples from astronomy, electrical engineering, remote sensing and medicine. It also shows how many, more traditional, methods can be enhanced by incorporating the wavelet and multiscale methods into the processing.

For graduate students and researchers already experienced in image processing and data analysis, this book provides a guide to a wide range of data-analysis techniques.

View/Download Image Processing and Data Analysis: The Multiscale Approach

 

Preface vii

 

 

1 The wavelet transform 1

 

1.1 Multiscale methods . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Some perspectives on the wavelet transform . . . . . . 2

1.1.2 The wavelet transform and the Fourier transform . . . 5

1.1.3 Applications of the wavelet transform . . . . . . . . . 7

1.2 The continuous wavelet transform . . . . . . . . . . . . . . . 8

1.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 The inverse transform . . . . . . . . . . . . . . . . . . 9

1.3 Examples of wavelet functions . . . . . . . . . . . . . . . . . . 9

1.3.1 Morlet’s wavelet . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Mexican hat . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.3 Haar wavelet . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 The discrete wavelet transform . . . . . . . . . . . . . . . . . 12

1.4.1 Multiresolution analysis . . . . . . . . . . . . . . . . . 13

1.4.2 Mallat’s horizontal and vertical analyses . . . . . . . . 16

1.4.3 Non-dyadic resolution factor . . . . . . . . . . . . . . 20

1.4.4 The `a trous algorithm . . . . . . . . . . . . . . . . . . 24

1.4.5 Pyramidal algorithm . . . . . . . . . . . . . . . . . . . 35

1.4.6 Scaling functions with a frequency cut-off . . . . . . . 39

1.4.7 Discussion of the wavelet transform . . . . . . . . . . 43

1.5 Multiresolution and median transform . . . . . . . . . . . . . 45

1.5.1 Multiresolution median transform . . . . . . . . . . . 45

1.5.2 Pyramidal median transform . . . . . . . . . . . . . . 46

1.5.3 Iterative pyramidal median transform . . . . . . . . . 47

1.5.4 Non-iterative pyramidal transform with exact reconstruction

. . . . . . . . . . . . . . . . . . . . . . . . . 47

1.5.5 Conclusion on multiscale median transforms . . . . . . 48

1.6 Multiresolution and mathematical morphology . . . . . . . . 48

1.6.1 Multiresolution . . . . . . . . . . . . . . . . . . . . . . 49

1.6.2 Pyramidal morphological transform . . . . . . . . . . 49

1.6.3 Conclusions on non-wavelet multiresolution approaches 50

2 Multiresolution support and filtering 51

2.1 Noise modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.1 Definition of significant coefficients . . . . . . . . . . . 51

2.1.2 Gaussian noise . . . . . . . . . . . . . . . . . . . . . . 52

2.1.3 Poisson noise . . . . . . . . . . . . . . . . . . . . . . . 53

2.1.4 Gaussian and Poisson noise . . . . . . . . . . . . . . . 55

2.1.5 Poisson noise with few photons or counts . . . . . . . 57

2.1.6 Other types of noise . . . . . . . . . . . . . . . . . . . 59

2.2 Multiresolution support . . . . . . . . . . . . . . . . . . . . . 60

2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.2.2 Multiresolution support from the wavelet transform . 61

2.2.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.2.4 Gaussian noise estimation from the multiresolution

support . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.2.5 Concluding remarks on the multiresolution support

and noise . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.3.1 Convolution using the continuous wavelet transform . 67

2.3.2 Wiener-like filtering in wavelet space . . . . . . . . . . 69

2.3.3 Hierarchical Wiener filtering . . . . . . . . . . . . . . . 70

2.3.4 Adaptive filtering . . . . . . . . . . . . . . . . . . . . . 72

2.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.4 Multiresolution image comparison . . . . . . . . . . . . . . . 82

3 Deconvolution 85

3.1 Introduction to deconvolution . . . . . . . . . . . . . . . . . . 85

3.2 Regularization using multiresolution support . . . . . . . . . 87

3.2.1 Noise suppression based on the wavelet transform . . . 87

3.2.2 Noise suppression based on the multiresolution support 88

3.2.3 Regularization of Van Cittert’s algorithm . . . . . . . 88

3.2.4 Regularization of the one-step gradient method . . . . 89

3.2.5 Regularization of the Richardson-Lucy algorithm . . . 89

3.2.6 Convergence . . . . . . . . . . . . . . . . . . . . . . . 89

3.2.7 Examples from astronomy . . . . . . . . . . . . . . . . 89

3.2.8 Conclusion on regularization using the multiresolution

support . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.3 Multiscale entropy and image restoration . . . . . . . . . . . 99

3.3.1 Image restoration using the maximum entropy method 100

3.3.2 Formalism of maximum entropy multiresolution . . . . 103

3.3.3 Deconvolution using multiscale entropy . . . . . . . . 105

3.3.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . 106

3.3.5 Another application of multiscale entropy: filtering . . 110

3.3.6 Conclusion on multiscale entropy and restoration . . . 111

3.4 Image restoration for aperture synthesis . . . . . . . . . . . . 112

3.4.1 Introduction to deconvolution in aperture synthesis . . 112

3.4.2 CLEAN and wavelets . . . . . . . . . . . . . . . . . . 115

3.4.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . 120

3.4.4 Observations of two evolved stars . . . . . . . . . . . . 121

3.4.5 Conclusion on interferometric data deconvolution . . . 126

4 1D signals and Euclidean data analysis 129

4.1 Analysis of 1D signals: spectral analysis . . . . . . . . . . . . 129

4.1.1 Spectral analysis . . . . . . . . . . . . . . . . . . . . . 129

4.1.2 Noise determination and detection criteria . . . . . . . 129

4.1.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . 130

4.1.4 Problems related to detection using the wavelet transform

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.1.5 Band extraction . . . . . . . . . . . . . . . . . . . . . 134

4.1.6 Continuum estimation . . . . . . . . . . . . . . . . . . 135

4.1.7 Optical depth . . . . . . . . . . . . . . . . . . . . . . . 138

4.1.8 The multiresolution spectral analysis algorithm . . . . 140

4.2 Wavelets and multivariate data analysis . . . . . . . . . . . . 142

4.2.1 Wavelet regression in multivariate data analysis . . . . 142

4.2.2 Degradation of distance through wavelet approximation144

4.2.3 Degradation of first eigenvalue through wavelet filtering146

4.3 The Kohonen map in wavelet space . . . . . . . . . . . . . . . 147

4.3.1 Example of SOFM in direct and in wavelet spaces . . 149

4.3.2 K-means and principal components analysis in wavelet

space . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.4 Multiresolution regression and forecasting . . . . . . . . . . . 155

4.4.1 Meteorological prediction using the `a trous method . . 155

4.4.2 Sunspot prediction using `a trous and neural networks 158

4.4.3 Dynamic recurrent neural network architecture . . . . 158

4.4.4 Combining neural network outputs . . . . . . . . . . . 159

5 Geometric registration 163

 

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