Stationary wavelet transform

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The Stationary wavelet transform (SWT) is similar to the DWT except the signal is never subsampled and instead the filters are upsampled at each level of decomposition.

Each level's filters are up-sampled versions of the previous.

The SWT is an inherently redundant scheme as each set of coefficients contains the same number of samples as the input – so for a decomposition of N levels there is a redundancy of 2N.

[edit] Synonyms

The idea of omitting the downsampling in the discrete wavelet transform is so obvious, 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
• Undecimated wavelet transform

Some of the names are disadvantageous. Decimation has an unfortunate military connotation. The SWT is not shift-invariant, because the result of the transform depends on the shift of the input. However it depends in a very simple way, that is shifting of the input yields shifting of the output. In contrast to the SWT the transform to the absolute Fourier spectrum is shift invariant. The SWT is clearly time-invariant, but interestingly the term "Time invariant wavelet transform" has not been established.

[edit] References

• James E. Fowler: The Redundant Discrete Wavelet Transform and Additive Noise, contains an overview of different names for this transform.