Image Processing Reference

In-Depth Information

∼
B
=

{

(

x, y

)

∈ O × O | φ
i
∈B φ
i
(

x

)

− φ
i
(

y

)

2
=0

}.

(8.7)

Tolerance Relations and Tolerance Classes

The concept of indiscernibility relation can be generalized to the tolerance rela-

tion, which is very important in near set theory (Peters, 2009, 2010; Peters and

Wasilewski, 2009; Peters and Ramanna, 2009; Peters and Puzio, 2009; Meghdadi,

Peters, and Ramanna, 2009). Tolerance relations emerge in transition from the con-

cept of
equality
to
almost equality
when comparing objects by they feature values.

Tolerance Relation

DEFINITION 8.4

ζ ⊆ X × X

X

A tolerance relation

in general, is a binary relation that is

reflexive and symmetric but not necessarily transitive (Sossinsky, 1986).

on a set

1.

ζ ⊂ X × X

,

2.

∀x ∈ X,

(

x, x

)

∈ ζ

,

3.

∀x, y ∈ X,

(

x, y

)

∈ ζ ⇒

(

y, x

)

∈ ζ

.

Moreover, the notation

xζy

can be used used as an abbreviation of (

x, y

)

∈ ζ

.

The set

X

supplied with the binary relation

ζ

is named a
tolerance space
and is

shown with

X
ζ
. The term tolerance space was originally coined by E. C. Zeeman

in (Zeeman, 1962).

A perceptual tolerance relation is defined in the context of perceptual systems as

follows, where
=
B,
is used instead of

ζ

, to denote the tolerance relation.

Perceptual Tolerance Relation

(Peters, 2009, 2010)

DEFINITION 8.5

Let
O,
F
be a perceptual system and let
∈
(set of all real numbers). For every

B⊆
F

the perceptual tolerance relation
=
B,ε
is defined as follows:

=
B,ε
=
{
(
x, y
)
∈ O × O
:
φ
B
(
x
)
−
φ
B
(
y
)
2
≤ ε},

(8.8)

)]
T
is a feature-value vector representing an

object description obtained using all of the probe functions in

where

φ
B
(

x

)=[

φ

1
(

x

)

φ

2
(

x

)

... φ
l
(

x

B

and

·
2
is the

L

2
norm (

L
p
norm in general (Janich, 1984)).

A tolerance relation =
B,ε
defines a covering on the set
O
of perceptual objects,

resulting in a set of tolerance classes and tolerance blocks (Bartol, Mir, Piro, and

Rossell, 2004; Schroeder and Wright, 1992). In this work, the definition of tolerance

class in (Bartol et al., 2004) has been used as follows and is similar to equivalence

class in (8.3).

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