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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