Image Processing Reference
In-Depth Information
TABLE 2.6: Functional forms of d(x;B)'s in 2-D [71], p = 1, 2, 3, 4. For
compactness we use α
1
= max(|x
1
|,|x
2
|),α
2
= |x
1
|+|x
2
|.
p Well-Behaved B d(x; B)
1
{1}
α
2
: Simple
{2}
α
1
: Simple
2
{1,2}
max(⌈2α
2
/3⌉,α
1
): Simple
3
{1,1,2}
max(⌈3α
2
/4⌉,α
1
): Simple
{1,2,2}
max(⌈3α
2
/5⌉,α
1
): Simple
4
{1,1,1,2}
max(⌈4α
2
/5⌉,α
1
): Simple
{1,1,2,2}
max(⌊(α
2
+ 1)/6⌋ + ⌊(α
2
+ 3)/6⌋ + ⌊(α
2
+ 4)/6⌋ + ⌊(α
2
+ 5)/6⌋,α
1
)
{1,2,1,2}
max(⌈2α
2
/3⌉,α
1
): Simple; same as d( 1, 2)
{1,2,2,2}
max(⌈4α
2
/7⌉,α
1
): Simple
Theorem 2.14. d(B) is a metric iff B is well-behaved.
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The general functional form of a d(B) is complex. However, it usually
takes a simple form when the corresponding d(B) is a metric, as the following
example illustrates in low dimensions.
Example 2.13. In Tables 2.6 and 2.7, we present the functional forms of
d(B)'s in 2-D and 3-D for short sequences.
Theorems 2.13 and 2.14 present strong characterizations for N-Sequence
distances in n-D that generalize the results for O(m)-neighbor distances in
d
m
and also generate a set of new metrics. Their further characterizations
in terms of hyperspheres are presented in Section 2.5.5 where we observe
that the hyperspheres of these distances asymptotically approach shapes of
hyperoctagons, justifying their nomenclature as hyperoctagonal distances.
2.4.2 Octagonal Distances in 2-D
Hyperoctagonal distances presented in the last section take interesting
forms when we restrict them to 2-D. These generalize the octagonal distance
originally proposed by Rosenfeld and Pfaltz [182].
First, we specialize Theorem 2.13 (from [56]) to get a simpler form for a
d(B).
Theorem 2.15. In 2-D, d(x;B) = |Π
∗
(x;B)| and is given by:
d(x;B) = max{d
1
(x),d
2
(x)}
where
p
j=1
x
1
+x
2
−f(j−1)−1
f(p)
d
1
(x)
= p +
d
2
(x)
=
max(x
1
,x
2
).
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