Image Processing Reference

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to relax the equivalence relations needed in mathematical world and extend it to indiscerni-

bility relations where almost solutions are possible.

In this section, a new similarity measure based on tolerance classes and near sets, for

comparison and analysis of images will be introduced. The new measure called tolerance

cardinality distribution nearness measure (TCD) is based on statistical distribution of the

size (cardinality) of the tolerance classes in each image. The idea behind it is that if images

are similar to each other, they should have corresponding tolerance classes with more or

less the same size and hence similar size distribution functions. The histogram comparison

approach in definition of TCD nearness measure was inspired by TOD nearness measure

as proposed in (Meghdadi, Peters, and Ramanna, 2009). However, TCD is fundamentally

different with TOD. While TCD calculates and compares distribution of the size of tolerance

classes, TOD just measures the overlap between tolerance classes and the size of tolerance

classes have no direct impact on TOD.

A similarity measure is proposed here based on statistical distribution of the size (cardi-

nality) of the tolerance classes in each image. The size of each tolerance class is defined as

the number of perceptual objects (subimages) in that tolerance class. Definition of TCD

is based on the basic idea that if images are similar to each other, they should have corre-

sponding tolerance classes with ”almost” the same size and hence similar size distribution

functions.

Let x
/
∼

denote tolerance classes in the coverings of X, Y defined by
∼
=
B,ε
. Next,

=
B,ε
, y
/
∼

=
B,ε

let c(x
/
∼

=
B,ε
) and c(y
/
∼

=
B,ε
) represent normalized cardinality of the tolerance classes x
/
∼

=
B,ε

and y
/
∼

=
B,ε
, respectively, i.e., cardinalities computed in (9.3) and (9.4).

=
B,ε
) =
x
/
∼

=
B,ε

|X|
,

c(x
/
∼

(9.3)

=
B,ε
) =
y
/
∼

=
B,ε

|Y|
.

c(y
/
∼

(9.4)

Suppose that{b
1
, b
2
, ..., b
N
b
}is a set of discrete bins for calculation of the histograms of

c(x
/
∼

=
B,ε
) where x∈X and y∈Y . The histogram (or empirical distribution

function) of c(x
/
∼

=
B,
) and c(y
/
∼

=
B,ε
) at bin value b
j
is shown as H
c
X
(b
j
) and defined as the number

of tolerance classes with the number of subimages (size) that belongs to j
th

bin.

The

cumulative distribution function is then defined as follows:

i

=

j

X

CH
c
X
(b
j
) =

H
c
X
(b
i
).

(9.5)

i=1

CH
c
Y
(b
j
) is similarly defined for image Y . The Tolerance Cardinality Distribution (TCD)

nearness measure is defined by taking the sum of differences between cumulative histograms

as defined in equation 9.6.

0

1

j=N
b

X

@

A
.

T CD = 1−

|CH
c
X
(b
j
)−CH
c
Y
(b
j
)|

(9.6)

j=1

The histogram comparison approach in definition of TCD nearness measure was inspired

by the TOD nearness measure as proposed in

(Meghdadi et al., 2009). However, TCD is

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