Digital Signal Processing Reference
In-Depth Information
2
H
( ) 48.6 ( 0.8491)( 0.7717)( 1.087)(
w @
× +
x
x
+
x
-
x
+
1.9934 0.994)
x
+
P
(77)
2
2
2
2
(
x
+
1.0797 0.318)(
x
+
x
-
0.3849 0.1766)(
x
+
x
-
1.2882 0.5314)(
x
+
x
-
1.9338 0.9726)
x
+
In order to obtain a filter with circular symmetry from the factorized 1D prototype function,
we replace in (12) cos
ω
with the circular cosine function (74). For instance, corresponding to
(12), the filter template A results in general as the discrete convolution:
A
= ×
A A
*
* *
K
A A A
*
*
* *
K
A
(78)
11
12
1
n
21
22
2
m
where
A
1
i
(
i
=1…
n
) are 3×3 templates and
A
2
j
(
j
=1…
m
) are 5×5 templates, given by:
A
1
i
=
C
+
a
i
⋅
A
01
and
A
2
j
=
C
∗
C
+
a
1
j
⋅
C
0
+
a
2
j
⋅
A
02
, where
A
01
is a 3×3 zero template and
A
02
a
5×5 zero template with the central element equal to one;
C
0
is a 5×5 template
obtained by bordering C with zeros. The above expressions correspond to the factors in (12).
The frequency response
H
C
(
ω
1
,
ω
2
) of the 2D circular filter results in a factorized form by
substituting
x
=
C
(
ω
1
,
ω
2
) in (77). Even if the filter results of high order, with very large
templates, next we show that using the Singular Value Decomposition (SVD), the resulted 2D
filter can be approximated with a negligible error. For the filter template B we can write
B
=
U
B
×
S
B
×
V
B
. The vector of singular values
S
B
of size 1×27 has 14 non-zero elements:
S
B
1
= 0.50536 0.086111 0.032794 0.013627 0.00521 0.002937 0.001935
0.001061 0.000639 0.000451 0.000418 0.0000385 0.0000196 0.00000144]
(a)
(b)
Figure 13.
Frequency response magnitude (a) and contour plot (b) of a circular FIR filter
Let us denote the vector above as
S
B
1
=
s
k
, with
k
=1...14 in our case. The exact filter matrix B
can be written as:
B
=
U
B
1
×
S
B
1
×
V
B
1
, where
U
B
1
and
V
B
1
are made up of the first 14 columns
of the unitary matrices
U
B
and
V
B
. If we consider the first largest M values of the vector
S
B
1
=
s
k
, the matrix B can be approximated as:
