Image Processing Reference
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topologically identical as long as the absolute difference vectors between them
are same. It then follows that for any distance function d,
d(u
1
,v
1
) = d(u
2
,v
2
), if |u
1
−v
1
| = |u
2
−v
2
|.
That is, the difference vector between any two points decides the distance
between them and the same is not dependent on the end point vectors. We
simplify the distance functions to its norm as follows:
Definition 2.13. ∀u,v ∈ Z
n
, norm d
N
: Z
n
→ R
+
∪{0} of a distance
function d : Z
n
×Z
n
→ R
+
∪{0} is defined as:
d
N
(x) = d(u,v) = d(0,u−v), where x = u−v
If d is symmetric, d(u,v) = d(v,u) and d
N
(x) = d
N
(−x) = d
N
(|x|).
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We use the distance function and its norm interchangeably in this chapter
and represent them by the same symbol whenever the context is clear.
2.2.4 Metricity Preserving Transforms
Throughout this chapter, we explore various neighborhoods and study the
metric property of their corresponding distance functions. In the process, we
often need to prove the metric property of distance functions and derive nec-
essary and su
cient conditions for their metricity. While these proofs can be
obtained independently in every case, we present a few generic results here
that help shorten many of the arguments.
We start with the observation that most distance functions presented here
are transformations or compositions of a number of simpler functions for which
the metric property is either already known or is easy to establish. So the
metricity of a composite distance is trivial if the transformation function sat-
isfies a set of properties. Mathematically, we pose the following question:
Let σ : R
+
∪{0}→ R
+
∪{0} be a transformation function and d : A×A →
R
+
∪ {0} be a metric (where A is Z
n
or R
n
or some other set). Is their
composition D = σ◦d a metric on A or not? Naturally,
D : A×A → R
+
and
D(x,y) = σ(d(x,y)) ∀x,y ∈A.
Examples of such transformations include:
Normalization of the range of d with two finite constant C
1
and C
2
such
that C
1
≤ D(x,y) ≤ C
2
,∀x,y ∈A, or
Approximation of d to the quantized space where the range of d is re-
stricted to the set of non-negative integers P.
Example 2.5. Table 2.2 shows examples of transformations for the Euclidean
distance E
2
: R
2
×R
2
→ R
+
∪{0} in 2-D.
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