0.4 Non-decimated wavelet transform

Suppose we have some signal $\left\{y_i\right\}$ observed at some equispaced design points : $y_i=f(i/n),i=1,...,n$ with $n=2^J,J\in \mathbb{N}$ . The transform presented in the previous section is sometimes called `decimated' because, for each scale $j$ , the coefficients $d_{jk}$ give only some information about the signal near the positions $x=2^{-j}k$ , and not near all the existing design points $2^{-J}k=k/n$ .

For this reason, the decimated wavelet transform lacks the property of translation invariance: given $t_0\in \mathbb{R}$ , the wavelet decomposition of $f(.)$ and of $f(.-t_0)$ are in general completely different. This may lead to some unwanted pseudo-Gibbs oscillations near a discontinuity which is not localized at a dyadic point [link] .

One remedy to this drawback consists in using a non-decimated wavelet transform (NDWT) , also called translation-invariant (TI) [link] or stationary [link] . The idea behind the NDWT is to a perform a discrete wavelet transform, not only of the original sequence ${\left\{y_i\right\}}_{i=1}^n$ , but of all the possible shifted sequences ${\left(S_{h}y\right)}_t=y_{(t+h)modn}$ . In terms of wavelet functions, this transform corresponds to a set of functions

At a given scale $j$ , the NDWT coefficients are thus present at all the locations $k/n$ for $k=1,...,n$ and give information about the signal at each observed design point. In other words, the non-decimated transform fills in the gap introduced in the decimated transform, see [link] .

Since we have $J-j_0$ scales and at each scales $n$ detail coefficients, the NDWT gives an overdetermined representation of the original signal ${\left\{y_i\right\}}_{i=1}^n$ and the wavelet coefficients $\{d_{jk},j=0,...,J-1,k=1,...,n\}$ are related to many different bases. Therefore the inverse operator will not be unique. A particular inverse, the averagebasis, corresponds to systematically average out the inverse wavelet transform obtained from each decimated wavelet transform that constitutes the translation-invariant transform. This makes the reconstruction robust with respect to a bad choice of a particular basis. Moreover, this average basis provides a smoother reconstruction than the original, decimated, transform [link] , [link] .

It allows for a (nearly) exact reconstruction of piecewise linear functions, instead of piecewise constant functions for the decimated Haar transform [link] .