Image Processing Reference
In-Depth Information
Next we verify the polynomials derived above.
||S(2,3; 2)|| = 74 = 2−4∗2 + 20∗2
2
= surf(2,3; 2)
||S(2,3; 1)|| = 18 = 2−4 + 20 = surf(2,3; 1)
||H(2,3; 2)|| = 1 +||S(2,3; 1)||+||S(2,3; 2)|| = 1 + 18 + 74
= 93 = 1 +
10
3
∗2 + 8∗2
2
+
20
3
∗2
3
= vol(2,3; 2),
||H(2,3; 1)|| = 1 +||S(2,3; 1)|| = 1 + 18
= 19 = 1 +
10
3
+ 8 +
20
3
= vol(2,3; 1).
€
TABLE 2.9: Volume and surface polynomials [70] for hyperspheres (n =
1,2,...,5).
Params
Coe
cients in surf(m,n;r)
Coe
cients in vol(m,n;r)
Euclidean
n m r
0
r
1
r
2
r
3
r
4
r
0
r
1
r
2
r
3
r
4
r
5
S
n
(r) V
n
(r)
1
1
2
-
-
-
-
1
2
-
-
-
-
0
2r
πr
2
2
1
0
4
-
-
-
1
2
2
-
-
-
2πr
2
2
0
8
-
-
-
1
4
4
-
-
-
8
3
4
3
4
3
πr
3
3
1
2
0
4
-
-
1
2
-
-
4πr
2
10
3
20
3
3
2
2
-4
20
-
-
1
8
-
-
3
3
2
0
24
-
-
1
6
12
8
-
-
16
3
8
3
8
3
10
3
4
3
2
3
2π
2
r
3
1
2
π
2
r
4
4
1
0
0
-
1
-
16
3
32
3
4
2
0
16
-16
32
-
1
8
8
-
32
3
184
3
50
3
46
3
4
3
0
-8
-
1
4
28
-
4
4
0
16
0
64
-
1
8
24
32
16
-
20
3
4
3
46
15
10
3
8
3
2
3
4
15
8
3
π
2
r
4
8
15
π
2
r
5
5
1
2
0
0
1
−32
3
−88
3
62
15
40
3
44
3
32
3
36
5
5
2
2
52
36
1
7
3
−176
3
46
5
50
3
142
3
124
5
5
3
2
60
124
1
32
196
3
476
3
18
5
80
3
212
3
232
3
476
15
5
4
2
-8
-8
1
5
5
2
0
80
0
160
1
10
40
80
80
32
Reprinted from
InformationSciences
, 50(1990), P. P. Das and B. N. Chatterji, Hyperspheres in Digital
Geometry, 73-91, Copyright (1990), with permission from Elsevier.
From Corollary 2.9 and the Table 2.9 we observe the following:
1. Similar to the Euclidean norm, the surface and volume expressions in
n-D digital space are polynomials in r of degree n−1 and n, respectively.
2. Unlike the Euclidean measures that involve π, the coe
cients in these
