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).
Search WWH ::

Custom Search