Pit Pattern Classification in Colonoscopy using Wavelets - WaveLab

2.3 Discrete wavelet transform
Using equation (2.2), T (a, b) can be rewritten in a more compact way as
T (a, b) =
∫ ∞
−∞
x(t)ψ a,b (t)dt (2.4)
This can than be expressed as an inner product of the signal x(t) and the wavelet function
ψ a,b (t)
T (a, b) = 〈x, ψ a,b 〉 (2.5)
Since a, b ∈ R this is called the continuous wavelet transform. Simply spoken equation
(2.5) returns the correlation between a signal x(t) and a wavelet ψ a,b .
2.3 Discrete wavelet transform
The discrete wavelet transform is very similar to the continuous wavelet transform, but while
in equation (2.2) the parameters a and b were continuous, in the discrete wavelet transform
these parameters are restricted to discrete values. To achieve this, equation (2.2) is slightly
modified
ψ m,n (t) =
=
(
1 t − nb0 a m 0
√ ψ a
m
0
a m 0
)
1
√ a
m
0
ψ ( a −m
0 t − nb 0
)
(2.6)
(2.7)
where m controls the dilation and n the translation with m, n ∈ Z. a 0 is a fixed dilation
step parameter greater than 1 and b 0 is the location parameter which must be greater than
0.
The wavelet transform of a continuous signal x(t) using discrete wavelets of the form of
equation (2.7) is then
T m,n = 1 √ a
m
0
∫ ∞
−∞
x(t)ψ ( a −m
0 t − nb 0
)
dt (2.8)
which again can be written in a more compact way
T m,n =
∫ ∞
−∞
x(t)ψ m,n (t)dt (2.9)
and therefore leads to
T m,n = 〈x, ψ m,n 〉 (2.10)
9