Digital Signal Processing Reference
In-Depth Information
3. Biorthogonal wavelets:
They have properties similar to those of the
orthogonal wavelets but are less restrictive.
4. Generalized filter bank representations:
They represent a general-
ization of the (bi)orthogonal wavelet packets. Each band is split into two
subbands. The basis functions fulfill the biorthonormality condition:
∞
g
c
(
m
−
n
c
l
)
h
k
(
n
k
n
−
m
)=
δ
(
c
−
k
)
δ
(
l
−
n
)
.
(2.88)
m=−∞
5. Oversampled wavelets:
There is no downsampling or oversampling
required, and
n
k
= 1 holds for all bands.
The first four wavelet types are known as
nonredundant wavelet
representations
. For the representation of oversampled wavelets, more
analysis functions (
) than basis functions are required. The
analysis and synthesis functions must fulfill
{
u
k
(
n
)
}
M−1
∞
g
k
(
m
−
l
)
h
k
(
n
−
m
)=
δ
(
l
−
n
)
.
(2.89)
k=0
m=−∞
This condition holds only in the case of linear dependency. This means
that some functions are represented as linear combinations of others.
2.6
The Two-Dimensional Discrete Wavelet Transform
ψ
j,n
}
(j,n)∈Z
2
in
L
2
(
R
), there also
exists a separable wavelet orthonormal basis in
L
2
(
R
):
For any wavelet orthonormal basis
{
{
ψ
j,n
(
x
)
ψ
l,m
(
y
)
}
(j,l,n,m)∈Z
4
(2.90)
The functions
ψ
j,n
(
x
)
ψ
l,m
(
y
) mix the information at two different scales
2
j
and 2
l
,across
x
and
y
. This technique leads to a building proce-
dure based on separable wavelets whose elements represent products of
function dilation at the same scale. These multiscale approximations
are mostly applied in image processing because they facilitate the pro-
cessing of images at several detail levels. Low-resolution images can be
represented using fewer pixels while preserving the features necessary
for recognition tasks.
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