Image Processing Reference
In-Depth Information
Theorem 2.2. A transformation σ
n
: (R
+
∪{}
n
) → R
+
∪{0} is metricity
preserving iff σ
n
satisfies the following conditions:
1. σ
n
is total.
2. σ
n
(x
1
,x
2
,...,x
n
) = 0 iff x
1
= x
2
= ··· = x
n
= 0.
∈ R
+
∪{0}, 1 ≤ i ≤ n, σ
n
(x
1
,x
2
,...,x
n
) + σ
n
(y
1
,y
2
,...,y
n
)
≥ max
x
i
+y
i
z
i
=|x
i
−y
i
|
3. ∀x
i
,y
i
{σ(z
1
,z
2
,...,z
n
)}.
1
≤i≤n
Proof. Su
ciency follows from Theorem 2.1. To prove necessity, assume A =
R
2
and d
1
= d
2
= d
3
= ··· = d
n
= E
2
. This leads to a similar contradiction
as in Theorem 2.1.
Example 2.8. σ
n
(d
1
,d
2
,...,d
n
) = d
1
+d
2
+···+d
n
and σ
n
(d
1
,d
2
,...,d
n
) =
max(d
1
,d
2
,...,d
n
) are GMPTs from Theorem 2.2. These are observed by
[182]. Yet another GMPT
1/2
n
n
·d
i
/
σ
n
(d
1
,d
2
,...,d
n
) =
m
i
m
i
,
m
i
> 0,1 ≤ i≤ n has been a popular choice for weighted Mahalanobis distance
[123].
€
Example 2.9. Consider the i
th
maximum component function f
i
(u) of vector
|u|, ∀u ∈ R
n
(Definition 2.4). Clearly
i
d
i
(u,v) =
f
j
(u−v)
j=1
is a metric over R
n
. This induces a class of generalized metrics
max
i=1
d(u,v;M) =
{d
i
(u,v)/m
i
}
over R
n
, where M = {m
i
: 1 ≤ i ≤ n,m
i
∈ R
+
} and 1 ≤ m
i
≤ n. The direct
proof for the metricity of d(M) is quite involved. But
σ
1
(x) = x/m and
σ
n
(x
1
,x
2
,...,x
n
) = max
i=1
x
i
being GMPT's, the metricity of d(M) immediately follows, once the metric-
ity of d
′
i
s are established.
€
In a number of similar situations, identification of proper GMPT may save
a lot of effort for the metricity proofs as we shall observe in this chapter.
