Optimal Differential Filter on Hexagonal Lattice - Computational Intelligence Paradigms in Advanced Pattern Classification - page 80

The equation for the orientation derivation (113) also has the same meaning as
(95). Filter p a + p c is equal to p b , therefore, it gives a perpendicular differential

value. The element values of the filter, p a − p c , shown in Figure Figure 8 actually

forms horizontal differential filter, however, this norm is not the same as that of
p b . Therefore (113) adopts the constant 1

√ 3to p a

p c to avoid an anisotropic
error. Hence (113) has the same accuracy as (95) because the only difference be-
tween them is caused by existence or non-existence of a redundancy. The Staunton
filters and the derived consistent gradient filters with radius of 1 thus have the same
accuracy.

/ −

Fig. 8 Composed horizontal
filter by p a − p c .

-1 1

-2 2

0

-1

1

8.5 Experiment and Results

Java 1.5 was used for the experiments and BigDecimal class, which provides fixed-
point calculation, was used to avoid calculation error. StrictMath class was used
for mathematical functions. Artificial images of 201 × 201 pixels on square lat-

tices and 201 233 pixels on hexagonal lattices were generated to get similar area

and shape. Pixels located at a distance less than 90 from the center of both lattices
were taken into consideration. To construct the artificial test images, the previously
defined functions with the following parameters were used: R 1

×

= 1000 for f ex 1 ,

A 2 for f ex 3 . The period

of f ex 2 is 4, and the period of f ex 3 is 4 at the boundary of the evaluated region
whose radius is 90. That is, to avoid any aliasing errors, only those artificial images
well under the Nyquist frequency were sampled. These test images are shown in
Figure 9.
The experimental results are summarized as follows. For each image, the error
statistics (mean, variance, maximum and minimum) are shown. Table 3 lists the
results for f ex 1 , Table 4 lists the results for f ex 2 , and Table 5 lists the results for f ex 3 .
Visual versions of these results are shown in Figure 10, Figure 11, and Figure 12.
The error is mapped to the luminance value. To show the difference in the accuracy
by four filters, the minimum and maximum errors are mapped to the luminance
values of 0 and 255. As stated above, only pixels located at a distance less than 90
from the center are included in the evaluation. Pixels outside this area are mapped
to a luminance value of 0.

= 255 , ω = π / 4for f ex 2 ,and A 3 = 255 , ω = π / ( 8 ∗ 90 )

2 3