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where l is the length of the description, and each φ i : O−→
inBis a probe function that
represents a feature used in the description of the object x (Pavel, 1983). Furthermore, a
contains all the probe functions used to describe an object x withB⊆
. Next, a
perceptual information system S can be defined as S =hO,
is the
set of all possible probe functions that take as the domain objects in O, and{V al φ i } φ i ∈F
,{V al φ i } φ i ∈F i, where
the value range of a function φ i
. For simplicity, a perceptual system is abbreviated as
iwhen the range of the probe functions is understood. It is the notion of a perceptual
system that is at the heart of the following definitions.
Definition 1 Perceptual Indiscernibility Relation (Peters, 2009c).
ibe a
perceptual system. For everyB⊆
the indiscernibility relation∼ B is defined as follows:
B ={(x,y)∈O×O :kφ B (x)−φ B k= 0},
wherek·krepresents the l 2 norm. IfB={φ}for some φ∈
, instead of writing∼ {φ} , we
write∼ φ .
Defn. 1 is a refinement of the original indiscernibility relation given by Pawlak in 1981 (Pawlak,
1981). Using the indiscernibility relation, objects with matching descriptions can be grouped
together forming granules of highest object resolution determined by the probe functions in
B. This gives rise to an elementary set (also called an equivalence class)
x /∼ B ={x 0 ∈O|x 0 B x},
defined as a set where all objects have the same description. Similarly, a quotient set is the
set of all elementary sets defined as
O /∼ B ={x /∼ B |x∈O}.
Defn. 1 provides the framework for comparisons of sets of objects by introducing a concept
of nearness within a perceptual system. Sets can be considered near each other when they
have “things” in common. In the context of near sets, the “things” can be quantified by
granules of a perceptual system, i.e., the elementary sets. The simplest example of nearness
between sets sharing “things” in common is the case when two sets have indiscernible
elements. This idea leads to the definition of a weak nearness relation.
Definition 2 Weak Nearness Relation (Peters, 2009c)
ibe a perceptual system and let X,Y⊆O. A set X is weakly near to a set Y
within the perceptual systemhO,
i(X./ F Y ) iff there are x∈X and y∈Y and there is
such that x∼ B y. In the case where sets X,Y are defined within the context of a
perceptual system as in Defn 2, then X,Y are weakly near each other.
An example of Defn. 2 is given in Fig. 7.2 where the grey lines represent equivalence classes.
The sets X and Y are weakly near each other in Fig. 7.2 because they both share objects
belonging to the same equivalence class.
7.2.1 Perceptual Tolerance relation
When dealing with perceptual objects (especially, components in images), it is sometimes
necessary to relax the equivalence condition of Defn. 1 to facilitate observation of associa-
tions in a perceptual system. This variation is called a perceptual tolerance relation that de-
fines yet another form of near sets (Peters and Ramanna, 2009; Peters, 2009b,c). A tolerance
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