Digital Signal Processing Reference
In-Depth Information
2
2
B
= ×
b
Q Q
*
- × *
a
P P
;
A
= ×
a
Q Q
*
- × *
a
P P
+ × × × *
j a a
P Q
(36)
2
0
j
j
j
j
2
0
j
j
j
j
1
j
j
The 2D transfer function for each biquad is complex. The characteristics of a fan-type filter
designed with this method and based on the prototype filter of order 4 given by (6)-(7) is shown
in Fig.6 (a), for the indicated parameters. As with the diamond-type filter analyzed in the
previous section, the fan-type filter characteristic features marginal linearity distortions which
can be corrected using a LP filter, similar with the correction filter used in Section 3.2 and
having the frequency characteristic shown in Fig. 3 (a).
(b)
(a)
Figure 7.
(a) Frequency response magnitude and (b) contour plot for the 2-quadrant fan filter
Two corrected fan-type filters with specified parameters have the magnitudes and contour
plots shown in Fig.6 (b), (c). The initial distortions have been eliminated. With the same
correction filter, we obtain the two-quadrant fan filter, shown in Fig. 7, by setting the aperture
angle
θ
=
π
/2 and orientation angle
φ
=
π
/4.
5. Very selective multidirectional IIR Filters
In this section a design method based on spectral transformations is proposed for another class
of 2D IIR filters, namely multi-directional filters. The design starts from an analog prototype
with specified parameters. Applying an appropriate frequency transformation to the 1D
transfer function, the desired 2D filter is directly obtained in a factorized form, like the filters
designed in the previous sections. For two-directional filters, an example is given of extracting
lines with two different orientations from a test image. The spectral transformation used in the
case of multi-directional filters is similar to the one presented in the previous section, derived
for fan-type filters and given by (34), (35). In this section the design of two-directional and
three-directional filters with specified orientation is detailed. The method can be easily
generalized to arbitrary multi-directional filters.
