Information Technology Reference
In-Depth Information
11.2
Wavelet Basis Functions
11.2.1
Wavelet Frames
The continuous time wavelet is defined at scale a and b as
p
a
x
b
1
a;b
.x/ D
(11.1)
˛
It will be shown that a continuous time signal f.x/ can be expressed as a series
expansion of discrete wavelet basis functions. The discrete wavelet has the form
[
2
,
187
,
191
,
225
]
x
nb
0
˛
0
˛
0
1
p
˛
0
m;n
.x/ D
(11.2)
The wavelet transform of a continuous signal f.x/ using discrete wavelets of the
form of Eq. (
11.2
) is given by
x
nb
0
˛
0
˛
0
dx
Z
C1
f.x/
1
T
m;n
D
p
˛
0
(11.3)
1
which can be also expressed as the inner product T
m;n
D< f.x/;
m;n
>.Forthe
discrete wavelet transform, the values T
m;n
are known as wavelet coefficients. To
determine how good the representation of a signal is in the wavelet space one can
use the theory of wavelet frames. The family of wavelet functions that constitute
a frame are such that the energy of the resulting wavelet coefficients lies within a
certain bounded range of the energy of the original signal
C1
X
C1
X
2
AE
jT
m;n
j
BE
(11.4)
mD1
nD1
where T
m;n
are the discrete wavelet coefficients, A and B are the frame bounds, and
E is the energy of the signal given by E D
R
C1
2
. The values
of the frame bounds depend on the parameters ˛
0
and b
0
chosen for the analysis and
the wavelet function used. If A D B, the frame is known as tight and has a simple
reconstruction formula given by the finite series
2
dt
Djjf.x/jj
1
jf.x/j
C1
X
C1
X
1
A
f.x/D
T
m;n
m;n
.x/
(11.5)
mD1
nD1
A tight frame with A D B>1is redundant, with A being a measure of the
redundancy. When A D B D 1 the wavelet family defined by the frame forms
