- Definition 1: K-Regularity
A unitary 2-band scaling filter is said to be
KK-regular if it has a
polynomial factor of the form
P
K
z
P
K
z
, with
Pz=1+z-12
P
z
1
z
2
.
H
0
z=1+z-12KQz
H
0
z
1
z
2
K
Q
z
(1)
Here a unitary filter is defined as a FIR filter with
coefficients
hn
h
n
from the basic recursive equation
φt=∑nnhn2φ2t−n
φ
t
n
n
h
n
2
φ
2
t
n
satisfying the admissibility conditions and
orthogonality conditions, i.e.,
∑nnhn=2
n
n
h
n
2
and
∑kkhkhk+2m=δm
k
k
h
k
h
k
2
m
δ
m
Any unitary filter is at least 1-regular according to the
above definition.
In what range can KK take its
value?
The admissibility conditions and orthogonality conditions take
N2+1
N
2
1
degrees of freedom out of the total
NN degrees of freedom in
designing the filter
hn
h
n
, where NN is the
length of the filter.
Since
KK is at least 1,
KK-regularity is equivalent to a
set of
K−1
K
1
linear constraints on the scaling filter, this
takes another
K−1
K
1
degrees of freedom. So we have
1≤K≤N2
1
K
N
2
.
KK-regularity describes the
flatness of
Hω
H
ω
, the Fourier transform of
hn
h
n
, at
ω=0
ω
0
and
ω=π
ω
. The flatness at these two locations indicates how
well the scaling filter performs as a low-pass filter, and is
related to the smoothness of corresponding scaling and wavelet
functions. Figure 1 shows the
frequency response of Daubechies scaling filters with increasing
regularities.
It can be proved that putting
KK-regularity on scaling filter is
equivalent to the following statements:
-
All moments of the wavelet filters are zero,
μ
1
k=0
μ
1
k
0
, for
k∈01…K−1
k
0
1
…
K
1
.
-
All moments of the wavelets are zero,
m
1
k=0
m
1
k
0
, for
k∈01…K−1
k
0
1
…
K
1
.
-
The partial moments of the scaling filter are equal for
k∈01…K−1
k
0
1
…
K
1
.
-
The frequency response of the scaling filter has a zero of
order KK at
ω=π
ω
.
-
The
k
th
k
th
derivative of the magnitude-squared frequency response of
the scaling filter is zero at
ω=0
ω
0
, for
k∈12…2K−1
k
1
2
…
2
K
1
.
-
All polynomial sequences up to degree
K−1
K
1
can be expressed as a linear combination of
shifted scaling filters.
-
All polynomials of degree up to
K−1
K
1
can be expressed as a linear combination of
shifted scaling functions at any scale.
Statement 1-3 relate the regularity of scaling filter to
vanishing wavelet moments and the moments of scaling filter.
Statement 4 and 5 describes the flatness of the frequency
response of the scaling filter. The last two statements show the
power of polynomial representation as a consequence of
KK-regularity.
Daubechies filters utilize all the remaining
N2−1
N
2
1
degrees of freedom to get maximum number of
vanishing wavelet moments, i.e.,
K=N2
K
N
2
for Daubechies filters.
Here is a LabVIEW VI showing the Daubechies
scaling and wavelet functions, along with the frequency response
of scaling and wavelet filters for
k∈23…14
k
2
3
…
14
.
