Image Processing Reference

In-Depth Information

fundamentally different from TOD. While TCD calculates and compares distribution of the

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

sizes of tolerance classes in an image covering defined by a tolerance relation has no direct

impact on TOD.

When the number of features (probe functions) used in a system increases, the overlaps

between tolerance classes becomes less and less. In this case, TOD will not work well, since

it is based on the distribution of overlaps between tolerance classes. TCD, on the other

hand, does not depend on overlaps between tolerance classes and is not hampered by large

number of features (probe functions).

Tolerance overlap distribution nearness measure(TOD), introduced by Meghdadi and Peters

in (Meghdadi et al., 2009), is based on statistical comparison between tolerance classes at

each subimage. TOD between two images (Meghdadi et al., 2009) is defined as follows:

Suppose X, Y∈O are two images (sets of perceptual objects), and X
/
∼

=
B,
are

the sets of all tolerance classes for image X and Y as defined in formulas (1.5) and (1.6).

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:

and Y
/
∼

=
B,

Ω
X
/
∼
=
B,ε
(x) ={z
/
∼

=
B,ε
∈X
/
∼

=
B,ε
| x∈z
/
∼

=
B,ε
}.

(9.7)

Consequently, the normalized number of tolerance classes in X
/
∼

which are overlapping

=
B,ε

at x is named as ω and defined as follows:

Ω
X
/
∼
=
B,ε
(x)

X
/
∼

=
B,ε

ω
X
/
∼
=
B,ε
(x) =

.

(9.8)

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
/
∼

and the notations

=
B,ε

Ω
Y
(y) and ω
Y
(y) for the set Y
/
∼

=
B,ε
. Let{b
1
, b
2
, ..., b
N
b
}be discrete bins in calculation

of histograms of ω
X
(x) and ω
Y
(y) where x∈X and y∈Y . Therefore, the empirical

distribution function (histogram) of ω
X
(x) at bin value b
j
is shown here as H
ω
X
(b
j
) and

defined as the number of subimages x with a value of ω
X
(x) that belongs to j
th

bin. The

cumulative distribution function is then defined as follows:

i

=

j

X

CH
ω
X
(b
j
) =

H
ω
X
(b
i
).

(9.9)

i=1

CH
ω
Y
(b
j
) is similarly defined for image Y . The Tolerance Overlap Distribution (TOD)

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

as defined in equation 9.10 where γ is a scaling factor.

0

1

γ

j=N
b

X

@

A

T OD = 1−

|CH
ω
X
(b
j
)−CH
ω
Y
(b
j
)|

.

(9.10)

j

=1

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