Image Processing Reference
In-Depth Information
Reprinted from
Information Sciences
59(1992), P. P. Das, J. Mukherjee and B. N. Chatterji, The t-Cost Distance
in Digital Geometry, 1-20, Copyright (1992), with permission from Elsevier.
FIGURE 2.9: HS
1
(2) or circle of d
1
, in 2-D for radius = 2. HS, extends to
infinity on all four directions.
for 2
n−1
≤ p ≤ 2
n
− 1 (as d
p
is a non-metric for 1 ≤ p ≤ 2
n−1
− 1; from
Lemma 2.11 these quantities would be unbounded). Unfortunately, there are
no general results for these in the literature. So only the results of 2- and 3-D
are presented here.
Lemma 2.12. In 2-D
4r
2
+ 4r + 1
S
2
(r)
=
8r
V
2
(r)
=
2r
2
+ 2r + 1
S
3
(r)
=
4r
V
3
(r)
=
and in 3-D,
24r
2
+ 2.
8r
3
+ 12r
2
+ 6r + 1
S
4
(r)
=
V
4
(r)
=
12r
2
+ 6, r odd
4r
3
+ 6r
2
+ 6r + 3
S
5
(r)
=
V
5
(r)
=
r odd
12r
2
+ 2, r even
4r
3
+ 6r
2
+ 6r + 1
=
=
r even
6r
2
2r
3
+ 3r
2
+ 2r
S
6
(r)
=
r odd
V
6
(r)
=
r odd
6r
2
+ 2, r even
2r
3
+ 3r
2
+ 2r + 1
=
=
r even
4r
2
+ 2.
4/3r
3
+ 2r
2
+ 8/3r + 1
S
7
(r)
=
V
7
(r)
=
S
p
(r)
=
1.
r = 0.
€
2.5.5 Hyperspheres of Hyperoctagonal Distances
In this section we explore the properties of the hyperspheres of hyperoc-
tagonal distances d(B) (Section 2.4.1). Hence, the neighborhood set is taken
