Stationary wavelet transform

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 ${\displaystyle 2^{(j-1)}}$ in the ${\displaystyle j}$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 Holschneider et al.[6]

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

KIT

Applications

A few applications of SWT are specified below.

• Signal denoising
• Pattern recognition
• Brain image classification [7]
• Pathological brain detection[8]

Synonyms

• 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)