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Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

Started by HannesF99 September 5, 2013
i am currently implementing the 'Undecimated Wavelet Transform' with the 'a
trous' algorithm. See e.g.
http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf,
section II-A.

As far as I understand, the implementation (in 1D) for the 'decomposition'
and 'reconstruction' phase is very simple, it boils down to the convolution
with a low-pass and high-pass filter (where for levels > 0, one has to add
zeros between the coefficients of the filters).

E.g. for the decomposition filter of the 'Bior1.3' wavelet, i have the
following filters for level 0, 1, 2 of size 6, 11 (6+5) and 16 (6 + 10):

level 0: [c1 c2 c3 c4 c5 c6]

level 1: [c1 0 c2 0 c3 0 c4 0 c5 0 c6]

level 2: [c1 0 0 c2 0 0 c3 0 0 c4 0 0 c5 0 0 c6]

I hope I inserted the zeros on the right places, if not please correct me.
The actual values of the coefficients c1, c2, c3, c4, c5, c6 can be found
at http://wavelets.pybytes.com/wavelet/bior1.3/.

My questions is now, what is the 'anchor' of this filter kernels, how can
the anchor be calculated in a general way (for decomposition/reconstruction
filters of different wavelet classes) ? I suppose its something like
'anchor_index = round_down(kernel_size / 2)', when anchor_index is
0-indexed, but i'm not sure.

Note that for implementing a convolution with a certain kernel, one has to
know always the 'anchor' of the kernel (the index of the coefficient in the
filter which is multiplied with the 'current' array element).

	 

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