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| [[File:Stationary wavelet transform lena.png|thumb|Haar Stationary Wavelet Transform of [[Lenna|Lena]]]]
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| {{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>
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| ==Implementation==
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| The following block diagram depicts the digital implementation of SWT.
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| [[Image:Wavelets - SWT Filter Bank.png|frame|none|A 3 level SWT filter bank]]
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| In the above diagram, filters in each level are up-sampled versions of the previous (see figure below).
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| [[Image:Wavelets - SWT Filters.png|frame|none|SWT filters]]
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| ==Applications==
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| A few applications of SWT are specified below.
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| * Signal denoising
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| * Pattern recognition
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| ==Synonyms==
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| * Stationary wavelet transform
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| * Redundant wavelet transform
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| * Algorithme à trous
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| * Quasi-continuous wavelet transform
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| * Translation invariant wavelet transform
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| * Shift invariant wavelet transform
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| * Cycle spinning
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| * Maximal overlap wavelet transform (MODWT)
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| * Undecimated wavelet transform (UWT)
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| ==References==
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| <references/>
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| [[Category:Wavelets]]
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