Information Technology Reference
In-Depth Information
an orthonormal basis. Even if A¤B a reconstruction formula of f.x/ can be
obtained in the form:
C1
X
C1
X
2
A C B
f
0
.x/ D
T
m;n
m;n
.x/
(11.6)
mD1
nD1
where f
0
.x/ is the reconstruction which differs from the original signal f.x/ by
an error which depends on the values of the frame bounds. The error becomes
acceptably small for practical purposes when the ratio B=A is near unity. The closer
this ratio is to unity, the tighter the frame.
11.2.2
Dyadic Grid Scaling and Orthonormal Wavelet
Transforms
The dyadic grid is perhaps the simplest and most efficient discretization for practical
purposes and lends itself to the construction of an orthonormal wavelet basis.
Substituting ˛
0
D 2 and b
0
D 1 into Eq. (
11.2
) the dyadic grid wavelet can be
written as
p
2
m
x
n2
m
1
m;n
D
(11.7)
2
m
or more compactly
m
2
.2
m
x n/
m;n
.t/ D 2
(11.8)
Discrete dyadic grid wavelets are commonly chosen to be orthonormal. These
wavelets are both orthogonal to each other and normalized to have unit energy. This
is expressed as
Z
C1
1 if m D m
0
; and n D n
0
0 otherwise
m;n
.x/
m
0
;n
0
.x/
dx
D
(11.9)
1
Thus, the products of each wavelet with all others in the same dyadic system are
zero. This also means that the information stored in a wavelet coefficient T
m;n
is
not repeated elsewhere and allows for the complete regeneration of the original
signal without redundancy. In addition to being orthogonal, orthonormal wavelets
are normalized to have unit energy. This can be seen from Eq. (
11.9
), as using
m D m
0
and n D n
0
the integral gives the energy of the wavelet function equal to
unity. Orthonormal wavelets have frame bounds A D B D 1 and the corresponding
wavelet family is an orthonormal basis. An orthonormal basis has components
which, in addition to being able to completely define the signal, are perpendicular
to each other.
