Image Processing Reference

In-Depth Information

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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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