Image Processing Reference
In-Depth Information
TABLE 2.7: Functional forms of d(x;B)'s in 3-D [71], p = 1, 2, 3. For
compactness, we use α
1
= max(|x
1
|,|x
2
|,|x
3
|),α
2
= max(|x
1
| + |x
2
|,|x
2
| +
|x
3
|,|x
3
|+|x
1
|),α
3
= |x
1
|+|x
2
|+|x
3
|.
p Well-Behaved B d(x; B)
1
{1}
α
3
{2}
max(⌊(α
3
+ 1)/2⌋,α
1
)
{3}
α
1
2
{1,2}
max(⌊(2α
3
+ 2)/3⌋,α
1
)
{1,3}
max(⌊(α
3
+ 3)/4⌋ + ⌊(α
3
+ 2)/4⌋, ⌊(2α
2
+ 2)/3⌋,α
1
)
{2,3}
max(⌊(2α
3
+ 4)/5⌋,α
1
)
3
{1,1,2}
max(⌊(3α
3
+ 3)/4⌋,α
1
)
{1,1,3}
max(⌊(α
3
+ 4)/5⌋ + ⌊(α
3
+ 3)/5⌋ + ⌊(α
3
+ 2)/5⌋, ⌊(3α
2
+ 3)/4⌋,α
1
)
{1,2,2}
max(⌊(3α
3
+ 4)/5⌋,α
1
)
{1,2,3}
max(⌊(α
3
+ 2)/3⌋ + ⌊(α
3
+ 4)/6⌋, ⌊(3α
2
+ 4)/5⌋,α
1
)
{1,3,3}
max(⌊(α
3
+ 6)/7⌋ + ⌊(2α
3
+ 4)/7⌋, ⌊(3α
2
+ 4)/5⌋,α
1
)
{2,2,3}
max(⌊(α
3
+ 6)/7⌋ + ⌊(α
3
+ 4)/7⌋ + ⌊(α
3
+ 2)/7⌋,α
1
)
{2,3,3}
max(⌊(α
3
+ 7)/8⌋ + ⌊(α
3
+ 5)/8⌋ + ⌊(α
3
+ 2)/8⌋,α
1
)
Distances obtained from the above expression are presented in Table 2.6.
Next we present a lemma from [56] that simplifies the well-behaved con-
dition for metricity.
Lemma 2.9. In 2-D an N-Sequence B is well-behaved iff
f(i + j), i + j ≤p,
f(p) + f(i + j −p), i + j > p.
f(i) + f(j) ≤
€
Consider B = {2,1}. Here f(1) = 2, but f(1) + f(1) = 4 > f(1 + 1) =
f(2) = 1. So {2,1} is not well-behaved and d({2,1}) is not a metric as was
concluded earlier.
All of these d(B) distances are called octagonal distances as their
disks, H(B;r) = {x : d(x;B) ≤ r} take the shape of octagons with ver-
tices at {(±r,±m(r)),(±m(r),±r)} where ∀x, 0 ≤ x ≤ m(r), d((x,r);B) =
d((r,x);B) = r and ∀x, x > m(r), d((x,r);B) = d((r,x);B) > r. The diagram
in Fig. 2.8 shows the geometric structure of m(r)'s.
From [56] we have
Theorem 2.16. For a well-behaved B (metric d(B))
m(r) = ⌊r/p⌋·(f(p)−p) + f(r mod p)−(r mod p)
€
Note that m(r) is a function of B, and in specific cases may degenerate to
0 or to r giving rise to diamonds or squares.
We present a generalization of this result in n-D in Section 2.5.5 on hy-
perspheres of hyperoctagonal distances.
