Image Processing Reference

In-Depth Information

IfB={φ}for some φ∈

F

, instead of∼
{φ}
we write∼
φ
.

Definition 9.3.2 Perceptual Weak Indiscernibility Relation

LethO,

F

ibe a perceptual system. For everyB⊆

F

the weak indiscernibility relation'
B

is defined as follows:

'
B
=
(x, y)∈O×O|∃φ
i
∈B
€
k4φk= 0
.

IfB={φ}for some φ∈

F

, instead of'
{φ}
we write'
φ
.

The set of all perceptual objects in O that are indiscernible to an object x∈O is called

an equivalence class and is shown as x
/∼
B
. Note that all the elements in an equivalence

class are indiscernible to each other.

Definition 9.3.3 Tolerance Relation

A tolerance relation ζ⊆X×X on a set X in general, is a binary relation that is reflexive

and symmetric but not necessarily transitive (Sossinsky, 1986).

1. ζ⊂X×X,

2.∀x∈X,

(x, x)∈ζ,

3.∀x, y∈X,

(x, y)∈ζ⇒(y, x)∈ζ.

The basic idea in a tolerance view of images is to replace the indiscernibility relation in

rough sets with a tolerance relation.

(9.2a) original image (Lake)

(9.2b) gray scale image

(9.2c) Covering,p=20

(9.2d) Covering,p=40

(9.2e) Covering,p=60

(9.2f) Covering,p=100

FIGURE 9.2:
(See color insert for Figure 9.2a and 9.2c)
Coverings with different window

sizes

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