Image Processing Reference
In-Depth Information
IfB={φ}for some φ∈
, instead of∼ {φ} we write∼ φ .
Definition 9.3.2 Perceptual Weak Indiscernibility Relation
ibe a perceptual system. For everyB⊆
the weak indiscernibility relation' B
is defined as follows:
' B = (x, y)∈O×O|∃φ i ∈B k4φk= 0 .
IfB={φ}for some φ∈
, 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,
(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
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