Image Processing Reference
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FIGURE 8.1: An image and its 9 average gray levels subimages
Perceptual Indiscernibility Relation
(Peters, 2010)
DEFINITION 8.2
Let
O, F
be a perceptual system. Let
φ ∈B
,
x, y ∈ O
and let
φ B (
x
)=(
φ
1 (
x
)
,...,φ i (
x
)
,...,φ L (
x
))
denote a description of object x containing feature values represented by φ i ∈B .A
perceptual indiscernibility relation
B is defined relative to
B
as follows
B =
{
(
x, y
)
∈ O × O | φ B (
x
)
φ B (
y
)
2 =0
},
(8.2)
where
· 2
denotes the
L
2 (Euclidean) norm. The set of all perceptual objects in
O
that are indiscernible relative to an object
x ∈ O
is called an equivalence class ,
denoted by
x /∼ B . This form of indiscernibility relation introduced in (Peters, 2009)
is a variation of the very useful relation introduced by Z. Pawlak in 1981 (Pawlak,
1981a). Note that all of the elements in
x /∼ B
have matching descriptions, i.e. , the
objects in x /∼ B
are indiscernible from each other. Then, by definition,
FIGURE 8.2: An example of an image partitioned into 961 subimages
∀x ∈ O,
x /∼ B =
{y ∈ O | y ∼ B x}.
(8.3)
The indiscernibility relation partitions the set
O
to form the quotient set
O /∼ B ,a
set of
x /∼ B .

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