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Stationary wavelet transform

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Definition

The basic discrete wavelet transform (DWT) algorithm is adapted to yield a stationary wavelet transform (SWT) which is independent of the origin. The approach of the SWT is simple, which is by applying suitable high-pass and low-pass filters to the data at each level, resulting in the generation of two sequences at the subsequent level. Without employment of downsampling techniques, the length of the new sequences is maintained to be the same as the original sequences. Rather than employing decimation similar to the standard wavelet transform which removes elements, the filters at each level are adjusted by augmenting them with zero-padding, as explained in the following:

Zx2j=xj,Zx2j+1=0 for all integers j

D0rH[r]=HD0r

D0rG[r]=GD0r

where Z is the operator that intersperses a given sequence with zeros, for all integers j.

D0r is the binary decimation operator

H[r] is a filter with weights h2r[r]j=hj and hk[r]=0 if k is not a multiple of 2r.

G[r] is a filter with weights g2r[r]j=hj and gk[r]=0 if k is not a multiple of 2r.

The design of the filters H[r] and G[r] involve of inserting a zero between every adjacent pair of elements in the filter H[r−1] and G[r−1] respectively.

The designation of aJ as the original sequence cJ is required before defining the stationary wavelet transform.

aj−1=H[J−j]aj, for j=J,J−1,…,1

bj−1=G[J−j]aj, for j=J,J−1,…,1

where aj=bj, given the length of aj is 2J

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