From Tolerance Near Sets to Perceptual Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 176

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