Lusin's theorem: Difference between revisions
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[[File:Stationary wavelet transform lena.png|thumb|Haar Stationary Wavelet Transform of [[Lenna|Lena]]]] | |||
{{Context|date=October 2009}}The '''Stationary wavelet transform''' (SWT)<ref>James E. Fowler: [http://ieeexplore.ieee.org/iel5/97/32130/01495429.pdf?arnumber=1495429 The Redundant Discrete Wavelet Transform and Additive Noise], contains an overview of different names for this transform.</ref> 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 <math>2^{(j-1)}</math> in the <math>j</math>th level of the algorithm.<ref>A.N. Akansu and Y. Liu, On Signal Decomposition Techniques, Optical Engineering, pp. 912-920, July 1991.</ref><ref>M.J. Shensa, The Discrete Wavelet Transform: Wedding the A Trous and Mallat Algorithms, IEEE Transaction on Signal Processing, Vol 40, No 10, Oct. 1992.</ref><ref>M.V. Tazebay and A.N. Akansu, Progressive Optimality in Hierarchical Filter Banks, Proc. IEEE International Conference on Image Processing (ICIP), Vol 1, pp. 825-829, Nov. 1994.</ref><ref>M.V. Tazebay and A.N. Akansu, Adaptive Subband Transforms in Time-Frequency Excisers for DSSS Communications Systems , IEEE Transaction on Signal Processing, Vol 43, No 11, pp. 2776-2782, Nov. 1995.</ref> 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.<ref>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.</ref> | |||
==Implementation== | |||
The following block diagram depicts the digital implementation of SWT. | |||
[[Image:Wavelets - SWT Filter Bank.png|frame|none|A 3 level SWT filter bank]] | |||
In the above diagram, filters in each level are up-sampled versions of the previous (see figure below). | |||
[[Image:Wavelets - SWT Filters.png|frame|none|SWT filters]] | |||
==Applications== | |||
A few applications of SWT are specified below. | |||
* Signal denoising | |||
* Pattern recognition | |||
==Synonyms== | |||
* 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) | |||
==References== | |||
<references/> | |||
[[Category:Wavelets]] | |||
Revision as of 03:09, 22 October 2013
My name is Jestine (34 years old) and my hobbies are Origami and Microscopy.
Here is my web site; http://Www.hostgator1centcoupon.info/ (support.file1.com)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 in the th level of the algorithm.[2][3][4][5] 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.[6]
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).
Applications
A few applications of SWT are specified below.
- Signal denoising
- Pattern recognition
Synonyms
- 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)
References
- ↑ James E. Fowler: The Redundant Discrete Wavelet Transform and Additive Noise, contains an overview of different names for this transform.
- ↑ A.N. Akansu and Y. Liu, On Signal Decomposition Techniques, Optical Engineering, pp. 912-920, July 1991.
- ↑ M.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.V. Tazebay and A.N. Akansu, Progressive Optimality in Hierarchical Filter Banks, Proc. IEEE International Conference on Image Processing (ICIP), Vol 1, pp. 825-829, Nov. 1994.
- ↑ M.V. Tazebay and A.N. Akansu, Adaptive Subband Transforms in Time-Frequency Excisers for DSSS Communications Systems , IEEE Transaction on Signal Processing, Vol 43, No 11, pp. 2776-2782, Nov. 1995.
- ↑ 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.