Book contents
- Frontmatter
- Contents
- List of contributors
- Preface to the paperback edition
- Preface to the first edition
- 0 A guided tour through the book
- 1 Wavelet analysis: a new tool in physics
- 2 The 2-D wavelet transform, physical applications and generalizations
- 3 Wavelets and astrophysical applications
- 4 Turbulence analysis, modelling and computing using wavelets
- 5 Wavelets and detection of coherent structures in fluid turbulence
- 6 Wavelets, non-linearity and turbulence in fusion plasmas
- 7 Transfers and fluxes of wind kinetic energy between orthogonal wavelet components during atmospheric blocking
- 8 Wavelets in atomic physics and in solid state physics
- 9 The thermodynamics of fractals revisited with wavelets
- 10 Wavelets in medicine and physiology
- 11 Wavelet dimensions and time evolution
- Index
2 - The 2-D wavelet transform, physical applications and generalizations
Published online by Cambridge University Press: 27 January 2010
- Frontmatter
- Contents
- List of contributors
- Preface to the paperback edition
- Preface to the first edition
- 0 A guided tour through the book
- 1 Wavelet analysis: a new tool in physics
- 2 The 2-D wavelet transform, physical applications and generalizations
- 3 Wavelets and astrophysical applications
- 4 Turbulence analysis, modelling and computing using wavelets
- 5 Wavelets and detection of coherent structures in fluid turbulence
- 6 Wavelets, non-linearity and turbulence in fusion plasmas
- 7 Transfers and fluxes of wind kinetic energy between orthogonal wavelet components during atmospheric blocking
- 8 Wavelets in atomic physics and in solid state physics
- 9 The thermodynamics of fractals revisited with wavelets
- 10 Wavelets in medicine and physiology
- 11 Wavelet dimensions and time evolution
- Index
Summary
Abstract
We begin with a short review of the 2-D continuous wavelet transform (CWT) and describe a number of physical applications. Then we discuss briefly the mathematical background, namely coherent states derived from group representations, and we show how it allows a straightforward extension to more general situations, such as higher dimensions, wavelets on the sphere or time-dependent wavelets. We conclude with a short outline of the 2- D discrete wavelet transform, some generalizations and a few physical applications.
Introduction
As we have seen in Chapter 1, both the continuous wavelet transform (CWT) and the discrete wavelet transform (DWT) may be extended to two dimensions. Here also, many applications have been developed, in various branches of physics and in image processing. As in the 1-D case, the CWT is better adapted to analysis, for instance the detection of specific features in an image. This is true, in particular, for oriented features, if one uses a wavelet which is directionally selective. On the other hand, the strong point of the DWT is data compression, notably in transmitting or reconstructing a 2-D signal after processing (e.g. denoising).
We will spend most of the present chapter discussing the 2-D CWT, for two reasons. First, it admits a number of interesting physical applications, that we will describe in Section 2.3. The second motivation is that its mathematical background, namely group representation theory (Section 2.4), suggests a straightforward extension to more general situations, such as wavelets in higher dimensions, or on manifolds (a sphere, for instance), or time- dependent wavelets, a promising tool for motion tracking (Section 2.5).
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- Chapter
- Information
- Wavelets in Physics , pp. 23 - 76Publisher: Cambridge University PressPrint publication year: 1999
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