Information Technology Reference
In-Depth Information
Using the dyadic grid wavelet of Eq. (
11.7
) the discrete wavelet transform is
defined as
Z
C1
T
m;n
D
x.t/
m;n
.x/
dt
(11.10)
1
By choosing an orthonormal wavelet basis
m;n
.x/ one can reconstruct the original
signal f.x/ in terms of the wavelet coefficients T
m;n
using the inverse discrete
wavelet transform:
C1
X
C1
X
f.x/D
T
m;n
m;n
.x/
(11.11)
mD1
nD1
requiring the summation over all integers m and n. In addition, the energy of the
signal can be expressed as
Z
C1
C1
X
C1
X
2
dx
D
2
jf.x/j
jT
m;n
j
(11.12)
1
mD1
nD1
11.2.3
The Scaling Function and the Multi-resolution
Representation
Orthonormal dyadic discrete wavelets are associated with scaling functions and their
dilation equations. The scaling function is associated with the smoothing of the
signal and has the same form as the wavelet
m;n
.x/ D 2
m=2
.2
m
x n/
(11.13)
The scaling functions have the property
Z
C1
0;0
.x/
dx
D 1
(11.14)
1
where
0;0
.x/ D .x/ is sometimes referred as the father scaling function or father
(mother) wavelet. The scaling function is orthogonal to translations of itself, but
not to dilations of itself. The scaling function can be convolved with the signal to
produce approximation coefficients as follows:
Z
C1
S
m;n
D
f.x/
m;n
.x/
dx
(11.15)
1
One can represent a signal f.x/using a combined series expansion using both the
approximation coefficients and the wavelet (detail) coefficients as follows:
