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Non-decimated wavelet transform

Module by: Veronique Delouille. E-mail the author

Suppose we have some signal {yi}{yi} observed at some equispaced design points : yi=f(i/n),i=1,...,nyi=f(i/n),i=1,...,n with n=2J,JNn=2J,JN. The transform presented in the previous section is sometimes called `decimated' because, for each scale jj, the coefficients djkdjk give only some information about the signal near the positions x=2-jkx=2-jk, and not near all the existing design points 2-Jk=k/n2-Jk=k/n.

For this reason, the decimated wavelet transform lacks the property of translation invariance: given t0Rt0R, the wavelet decomposition of f(.)f(.) and of f(.-t0)f(.-t0) 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 [2].

One remedy to this drawback consists in using a non-decimated wavelet transform (NDWT) , also called translation-invariant (TI) [2] or stationary [3]. The idea behind the NDWT is to a perform a discrete wavelet transform, not only of the original sequence {yi}i=1n{yi}i=1n, but of all the possible shifted sequences (Shy)t=y(t+h)modn(Shy)t=y(t+h)modn. In terms of wavelet functions, this transform corresponds to a set of functions

ψ ˜ j k ( x ) = ψ ˜ ( 2 j ( x - 2 - J k ) ) , j = j 0 , ... , J - 1 , k = 0 , ... , 2 J - 1 . ψ ˜ j k ( x ) = ψ ˜ ( 2 j ( x - 2 - J k ) ) , j = j 0 , ... , J - 1 , k = 0 , ... , 2 J - 1 .
(1)

At a given scale jj, the NDWT coefficients are thus present at all the locations k/nk/n for k=1,...,nk=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 Figure 1.

Figure 1: Schema illustrating the translation-invariant version of the Haar transform. The points marked by are the one computed for the decimated Haar transform. At level JJ, one circulant shift is performed: the first observation is put at the end of the observed signal, and a second decimated transform is performed on the shifted data (yielding the points marked by at level J-1J-1). This process is iterated at the coarser levels, producing detail coefficients at all the points.
Figure 1 (image2.png)

Since we have J-j0J-j0 scales and at each scales nn detail coefficients, the NDWT gives an overdetermined representation of the original signal {yi}i=1n{yi}i=1n and the wavelet coefficients {djk,j=0,...,J-1,k=1,...,n}{djk,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 average basis, 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 [2], [1].

It allows for a (nearly) exact reconstruction of piecewise linear functions, instead of piecewise constant functions for the decimated Haar transform [2].

References

  1. Berkner, K. and Wells, R. (2002). Smoothness estimates for soft-threshold denoising via translation invariant wavelet transforms. Appl. Comput. Harmon. Anal., 12, 1-24.
  2. Coifman, R.R. and Donoho, D.L. (1995). Translation-Invariant De-Noising. In Antoniadis, A. and Oppenheim, G. (Eds.), Wavelets in Statistics, Lectures Notes in Statistics, Vol. 103. (pp. 125-150). Springer-Verlag.
  3. Nason, G.P. and Silverman, B.W. (1995). The stationary wavelet transform and some statistical applications. In Antoniadis, A. and Oppenheim, G. (Eds.), Wavelets in Statistics, Lectures Notes in Statistics, Vol. 103. (pp. 281-299). Springer-Verlag.

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