Discovering Image Similarities.Tolerance Near Set Approach - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 235

j . Notice, also, that
we can formulate a relationship between the description-based intersection

i

Let i, j∈I X , I Y be distinct indices used to identify the classes C , C

i T ∼ j

C

= B C

and

the product of the class sizes C

i · C j .

Proposition 1

X

i \
∼

≤ X
i∈I X ,
j∈I Y

C

i · C j .

X Y

j

X Y

C

= B,ε C

i∈I X ,
j∈I Y

A measure of the resemblance (i.e., image nearness measure N M ∼

= B (X, Y )) between a pair

of images X, Y is then formulated in Def. 6.

Definition 6 Tolerance Class Intersection-Based Nearness Measure (Peters and
Henry, 2009)
LethO, ibe a perceptual system and assume X, Y⊆O with coverings defined using the

tolerance relation ∼ = B .

F

j denote tolerance classes in X / ∼ = B , Y / ∼ = B , respectively.
Then a tolerance class intersection-based nearness measure (tiN M ∼

i

Let C , C

= B (X, Y )) is given in

(12.9) to measure the degree of nearness of X, Y is computed using

C

P

i T ∼ j

= B,ε C

i∈I X ,
j∈I Y

C

i · C j

tiN M ∼

= B,ε (X, Y ) =

P . (12.9)

i∈I X ,
j∈I Y

12.3.2 Tolerance Class Overlap Distribution Nearness Measure (Megh-
dadi et al., 2009)
A tolerance class overlap distribution (TOD) nearness measure is based on statistical com-
parison of overlaps between tolerance classes at each subimage. The method is as follows:
Suppose X, Y∈O are two images (sets of perceptual objects). The sets of all tolerance
classes for images X and Y are shown as follows and form a covering for each image.

X /∼ B,ε ={x /∼ B,ε | x∈X}, (12.10)

Y /∼ B,ε ={y /∼ B,ε | y∈Y}. (12.11)

Subsequently, the set of all overlapping tolerance classes corresponding to each object
(subimage) x is named as Ω X /∼ B,ε (x) and is defined as follows:

Ω X /∼ B,ε (x) ={z /∼ B,ε ∈X /∼ B,ε | x∈z /∼ B,ε }. (12.12)

Consequently, the normalized number of tolerance classes ω in X /∼ B,ε which are overlap-

ping at x is defined as follows:

Ω X /∼ B,ε (x)
X /∼ B,ε

ω X /∼ B,ε (x) = . (12.13)

Similarly, the set of all overlapping tolerance classes at every subimage y∈Y is denoted by
ω Y /∼ B,ε (y). Assuming that the set of probe functionsBand the value of are known, we use
the more simplified notation of Ω X (x) and ω X (x) for the set X /∼ B,ε

and the notations Ω Y (y)