For many years, digital signal processing has been governed by the theory of Fourier transform and its numerical implementation. The main disadvantage of Fourier theory is the underlying assumption that the signals have time-wise or space-wise invariant statistical properties. In many applications the deviation from a stationary behavior is precisely the information to be extracted from the signals. Wavelets were developed to serve the purpose of analysing such instationary signals. The book gives an introduction to wavelet theory both in the continuous and the discrete case. After developing the theoretical fundament, typical examples of wavelet analysis in the Geosciences are presented. The book has developed from a graduate course held at The University of Calgary and is directed to graduate students who are interested in digital signal processing. The reader is assumed to have a mathematical background on the graduate level.
Author(s): Wolfgang Keller
Publisher: Walter de Gruyter
Year: 2004
Language: English
Pages: 289
City: Berlin; w York
Preface......Page 4
Contents......Page 6
Notation......Page 8
1.1 Fourier analysis......Page 10
1.2 Linear filters......Page 23
2 Wavelets......Page 32
2.1 Motivation......Page 33
2.2 Continuous wavelet transformation......Page 39
2.3 Discrete wavelet transformation......Page 49
2.4 Multi-resolution analysis......Page 52
2.5 Mallat algorithm......Page 64
2.6 Wavelet packages......Page 72
2.7 Biorthogonal wavelets......Page 77
2.8 Compactly supported orthogonal wavelets......Page 91
2.9 Wavelet bases on an interval......Page 107
2.10 Two-dimensional wavelets......Page 111
2.11 Wavelets on a sphere......Page 119
3.1 Pattern recognition......Page 140
3.2 Data compression and denoising......Page 165
3.3 Sub-band coding, filtering and prediction......Page 190
3.4 Operator approximation......Page 205
3.5 Gravity field modelling......Page 221
3.1 Pattern recognition......Page 139
3.3 Sub-band coding, filtering and prediction......Page 189
A.1 Definition of Hilbert spaces......Page 226
A.2 Complete orthonormal systems in Hilbert spaces......Page 231
A.3 Linear functionals – dual space......Page 234
A.4 Examples of Hilbert spaces......Page 235
A.5 Linear operators – Galerkin method......Page 243
A.6 Hilbert space valued random variables......Page 245
B Distributions......Page 247
Exercises......Page 254
Bibliography......Page 278
Index......Page 286
