Module rgsl::types::wavelet_transforms [−][src]
Expand description
Wavelet Transforms
This chapter describes functions for performing Discrete Wavelet Transforms (DWTs). The library includes wavelets for real data in both one and two dimensions.
Definitions
The continuous wavelet transform and its inverse are defined by the relations,
w(s,\tau) = \int f(t) * \psi^*_{s,\tau}(t) dt and,
f(t) = \int \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau ds where the basis functions \psi_{s,\tau} are obtained by scaling and translation from a single function, referred to as the mother wavelet.
The discrete version of the wavelet transform acts on equally-spaced samples, with fixed scaling and translation steps (s, \tau). The frequency and time axes are sampled dyadically on scales of 2^j through a level parameter j. The resulting family of functions {\psi_{j,n}} constitutes an orthonormal basis for square-integrable signals.
The discrete wavelet transform is an O(N) algorithm, and is also referred to as the fast wavelet transform.
References and Further Reading
The mathematical background to wavelet transforms is covered in the original lectures by Daubechies,
Ingrid Daubechies. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics (1992), SIAM, ISBN 0898712742. An easy to read introduction to the subject with an emphasis on the application of the wavelet transform in various branches of science is,
Paul S. Addison. The Illustrated Wavelet Transform Handbook. Institute of Physics Publishing (2002), ISBN 0750306920. For extensive coverage of signal analysis by wavelets, wavelet packets and local cosine bases see,
S. G. Mallat. A wavelet tour of signal processing (Second edition). Academic Press (1999), ISBN 012466606X. The concept of multiresolution analysis underlying the wavelet transform is described in,
S. G. Mallat. Multiresolution Approximations and Wavelet Orthonormal Bases of L^2(R). Transactions of the American Mathematical Society, 315(1), 1989, 69–87. S. G. Mallat. A Theory for Multiresolution Signal Decomposition—The Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 1989, 674–693. The coefficients for the individual wavelet families implemented by the library can be found in the following papers,
I. Daubechies. Orthonormal Bases of Compactly Supported Wavelets. Communications on Pure and Applied Mathematics, 41 (1988) 909–996. A. Cohen, I. Daubechies, and J.-C. Feauveau. Biorthogonal Bases of Compactly Supported Wavelets. Communications on Pure and Applied Mathematics, 45 (1992) 485–560. The PhysioNet archive of physiological datasets can be found online at http://www.physionet.org/ and is described in the following paper,
Goldberger et al. PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation 101(23):e215-e220 2000.
Structs
The Wavelet structure contains the filter coefficients defining the wavelet and any associated offset parameters.
The centered forms of the wavelets align the coefficients of the various sub-bands on edges. Thus the resulting visualization of the coefficients of the wavelet transform in the phase plane is easier to understand.
The WaveletWorkspace structure contains scratch space of the same size as the input data and is used to hold intermediate results during the transform.