Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

2 Wavelets
Now, that we have defined the scaling function φ, we can construct a suited wavelet function
ψ m+1,n = √ 1 ∑
b k φ m,2n+k (t) (2.24)
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k
where
b k = (−1) k c Nk −1−k (2.25)
and N k is the number of scaling coefficients.
Analogous to equation (2.21) we can express b k as
b k = 〈φ(2t − k), ψ(t)〉 (2.26)
From equation (2.24) we can see, that the wavelet function ψ can be expressed in terms
of the scaling function φ. This is an important relationship which is used in the next section
to obtain the filter coefficients.
2.4 Filter based wavelet transform
In signal processing usually a signal is a discrete sequence. To analyze such a signal with
the wavelet transform based on filters so-called filter banks are needed, which guarantee a
perfect reconstruction of the signal. A filter bank consist of a low pass filter and a high
pass filter. While the low pass filter (commonly denoted by h) constructs an approximation
subband for the original signal, the high pass filter (commonly denoted by g) constructs a
detail subband consisting of those details, which would get lost if only the approximation
subband would be used for signal reconstruction.
To construct a filter bank we need to compute the coefficients for the low pass filter h first.
However, since the scaling equation is used to get the approximation for a signal, h can be
composed by using the coefficients c k from equation (2.20):
h[k] = c k (2.27)
Now, using equations (2.24) and (2.25), we are able compute g from h:
g[k] = (−1) k h[N k − 1 − k] (2.28)
For a discrete signal sequence x(t) the decomposition can be expressed in terms of a
convolution as
y a (k) = (h ∗ x)[k] (2.29)
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