Image Processing Reference
In-Depth Information
2.4.3 Weighted t-Cost Distance
In Section 2.3.2 it was shown that the t-cost distance D
t
(u) is a metric
in Z
n
. Further, it was presented in Section 2.3.2.4 that it is also a metric in
R
n
. In this section it is generalized by composing various t-cost norms with
a set of positive weights. This is known as the weighted t-cost norm [150]
WD
n
(x,y; w) : R
n
×R
n
→ R
+
∪{0} and is defined as follows:
Definition 2.28. ∀n,n≥ 1 and ∀x,y ∈ R
n
WD
n
(u; w) = max
1≤t≤n
{w
t
.D
t
(u)}
∈ R
+
,1 ≤ t ≤ n}. That is, w
t
's are
where u = |x − y| and w = {w
t
: w
t
non-negative real constants.
€
It may be noted that distances computed by WD
n
(u) are real-valued. A
modification to its form for integral distance values could be made in various
ways like:
WD
n
(u; w) = max
1≤t≤n
{⌈w
t
.D
t
(u)⌉}
The metric property of WD
n
(u; w) is given in the following theorem.
Theorem 2.18. WD
n
(u; w) is a metric in n-D real space R
n
.
Proof. The proof follows from the facts that every D
t
(u) is a metric, weights
w
t
's are non-negative and the composing function max is an MPT.
Corollary 2.7. WD
n
(u; w) is a metric in n-D real space R
n
.
€
D
n
(u)
m
As an m-neighbor norm, d
m
(u) = max
D
1
(u),
, WD
n
special-
izes to d
m
for w
1
= 1, w
n
=
1
m
and all other weights being zero. Also it
becomes the usual t-cost norm D
t
, when w
t
= 1 and all other weights are
zero. Hence, WD
n
provides a generalization with infinite possibilities includ-
ing both d
m
and D
t
norms.
In [167], d
m
and D
t
are generalized by the class of t-cost-m-Neighbor
distance. WD
n
presents another perspective for generalizing them.
2.5 Hyperspheres of Digital Distances
In the last section we studied a number of distance functions in n-D for var-
ious neighborhood set notions. Conditions for their metricity were established
and the relationships to shortest paths elucidated. We have also observed that
many of them preserve a certain order of their computed distance values.
The length of the shortest path, however, provides only one type of measure
