The Stationary wavelet transform (SWT) is a wavelet transform algorithm designed to overcome the lack of translation-invariance of the discrete wavelet transform

Discrete wavelet transform

In numerical analysis and functional analysis, a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled...

(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. 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.

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

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)