Stationary wavelet transform
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This article provides insufficient context for those unfamiliar with the subject. Please help improve the article with a good introductory style. (October 2009) |
The Stationary wavelet transform (SWT)[1] is a wavelet transform algorithm designed to overcome the lack of translation-invariance of the discrete wavelet transform (DWT). Translation-invariance is achieved by removing the downsamplers and upsamplers in the DWT and upsampling the filter coefficients by a factor of 2(j − 1) in the jth level of the algorithm[2]. The SWT is an inherently redundant scheme as the output of each level of SWT contains the same number of samples as the input – so for a decomposition of N levels there is a redundancy of N in the wavelet coefficients. This algorithm is more famously known as "algorithme à trous" in French (word trous means holes in English) which refers to inserting zeros in the filters. It was introduced by Holdschneider et al.[3]
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[edit] Implementation
The following block diagram depicts the digital implementation of SWT.
In the above diagram, filters in each level are up-sampled versions of the previous (see figure below).
[edit] Applications
A few applications of SWT are specified below.
- Signal denoising
- Pattern recognition
[edit] Synonyms
The idea of omitting the downsampling in the discrete wavelet transform is sufficiently intuitive that this variant was invented several times with different names.
- Stationary wavelet transform
- Redundant wavelet transform
- Algorithme à trous
- Quasi-continuous wavelet transform
- Translation invariant wavelet transform
- Shift invariant wavelet transform
- Cycle spinning
- Maximal overlap wavelet transform (MODWT)
- Undecimated wavelet transform (UWT)
[edit] References
- ^ James E. Fowler: The Redundant Discrete Wavelet Transform and Additive Noise, contains an overview of different names for this transform.
- ^ Mark J. Shensa, The Discrete Wavelet Transform: Wedding the A Trous and Mallat Algorithms, IEEE Transaction on Signal Processing, Vol 40, No 10, Oct. 1992.
- ^ M. Holschneider, R. Kronland-Martinet, J. Morlet and P. Tchamitchian. A real-time algorithm for signal analysis with the help of the wavelet transform. In Wavelets, Time-Frequency Methods and Phase Space, pp. 289–297. Springer-Verlag, 1989.
