Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

2.4 Filter based wavelet transform
and
y d (k) = (g ∗ x)[k] (2.30)
where y a and y d denote the approximation and the detail subband.
The discrete convolution between a discrete signal x(t) and a filter f(t) is defined as
(f ∗ x)(k) =
l∑
f(i)x(k − i) (2.31)
i=0
where l is the length of the respective filter f.
To avoid redundant data in the decomposed subbands, the signal of length N is downsampled
to length N/2. Therefore the result of equation (2.29) and (2.30) are sequences of
length N/2 and the decomposed signal has a length of N.
To reconstruct the original signal x from y a and y d a reconstruction filter bank consisting
of the filters ĥ and ĝ, which are the low pass and high pass reconstruction filters, is used.
Since during the decomposition the signal was downsampled, the detail and the approximation
subband have to be upsampled from size N/2 to size N. Then the following formula is
used to reconstruct the original signal x.
x(t) = (ĥ ∗ y a)(t) + (ĝ ∗ y d )(t) (2.32)
The reconstruction filters ĥ and ĝ can be computed from the previously computed decomposition
filters h and g using the following equations
ĥ[k] = (−1) k+1 g[k] (2.33)
and
ĝ[k] = (−1) k h[k] (2.34)
A special wavelet decomposition method without any downsampling and upsampling involved
is the algorithme à trous. This algorithm provides approximate shift-invariance with
an acceptable level of redundancy. Since in the discrete wavelet transform at each decomposition
level every second coefficient is discarded, this can result in considerably large
shift-variances.
When using the algorithme à trous however the subbands are not downsampled during
decomposition and the same amount of coefficients is stored for each subband - no matter
at which level of decomposition a subband is located in the decomposition tree. As a
consequence no upsampling is needed at all when reconstructing a signal.
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