Stationary wavelet transform

Haar Stationary Wavelet Transform of Lena

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]

Implementation

The following block diagram depicts the digital implementation of SWT.

A 3 level SWT filter bank

In the above diagram, filters in each level are up-sampled versions of the previous (see figure below).

SWT filters

Applications

A few applications of SWT are specified below.

• Signal denoising
• Pattern recognition

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)

References

1. ^ James E. Fowler: The Redundant Discrete Wavelet Transform and Additive Noise, contains an overview of different names for this transform.
2. ^ 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.
3. ^ 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.