Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

2.3 Discrete wavelet transform
From this equation it can be seen that the original signal is expressed as a combination of
an approximation of itself (at an arbitrary scale index m 0 ), added to a succession of signal
details from scales m 0 down to −∞. The signal detail at scale m is therefore defined as
d m (t) =
∞∑
n=−∞
T m,n ψ m,n (t) (2.18)
hence equation (2.17) can be rewritten as
x(t) = x m0 (t) +
∑m 0
m=−∞
d m (t) (2.19)
which shows that if the signal detail at an arbitrary scale m is added to the approximation
at that scale it results in an signal approximation at an increased resolution (m − 1).
The following scaling equation describes the scaling function φ(t) in terms of contracted
and shifted versions of itself:
φ(t) = ∑ k
c k φ(2t − k) (2.20)
where φ(2t − k) is a contracted version of φ(t) shifted along the time axis by step k ∈ Z
and factored by an associated scaling coefficient, c k , with
c k = 〈φ(2t − k), φ(t)〉 (2.21)
Equation (2.20) basically shows that a scaling function at one scale can be constructed
from a number of scaling functions at the previous scale.
From equation (2.13) and (2.20) and examining the wavelet at scale index m + 1, one can
see that for arbitrary integer values of m the following is true:
( )
2 − m+1 t
2 φ
2 − n = 2 − m m+1 2 2
− 1 2
∑
k
( )
2t
c k φ
2 · 2 − 2n − k m
(2.22)
which can be written more compactly as
φ m+1,n = √ 1 ∑
c k φ m,2n+k (t) (2.23)
2
k
That is, the scaling function at an arbitrary scale is composed of a sequence of shifted scaling
functions at the next smaller scale each factored by their respective scaling coefficients.
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