Image Processing Reference

In-Depth Information

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

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)

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