Image Processing Reference
InDepth Information
8.3.1
Tolerance Overlap Distribution nearness measure (TOD)
A similarity measure is proposed here based on statistical comparison of overlaps
between tolerance classes at each subimage. The proposed method is as follows.
Suppose
X, Y ∈ O
are two images (sets of perceptual objects).
The sets of all
tolerance classes for image
X
and
Y
are shown as follow and form a covering for
each image.
X
/
=
B,ε
=
{x
/
=
B,ε
 x ∈ X}
(8.17)
Y
/
=
B,ε
=
{y
/
=
B,ε
 y ∈ Y }
(8.18)
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
}
(8.19)
Consequently, the normalized number of tolerance classes in
X
/
=
B,ε
which are
overlapping at
x
is named as
ω
anddefinedasfollow:
)
Ω
X
/
=
B,ε
(
x
X
/
=
B,ε
ω
X
/
=
B,ε
(
x
)=
(8.20)
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 functions
B
and 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
)and
ω
Y
(
y
) for the set
Y
/
=
B,ε
.Let
{b
1
,b
2
, ..., b
N
b
}
are the discrete bins in calculation of histograms of
x ∈ X
and
y ∈ Y
. Therefore, the empirical distribution function (histogram) of
ω
X
(
x
)at
bin value
ω
X
(
x
)and
ω
Y
(
y
) where
x
with a value of
ω
X
(
x
) that belongs to
j
th
bin. The cumulative distribution function
is then defined as follows:
b
j
is shown here as
H
ω
X
(
b
j
) and defined as the number of subimages
i
=
j
i
=1
H
ω
X
(
CH
ω
X
(
b
j
)=
b
i
)
(8.21)
.The
Tolerance Overlap Distribution
(TOD)
nearness measure is defined by taking the sum of differences between cumu
lative histograms as defined in equation 8.22 where
CH
ω
Y
(
b
j
) is similarly defined for image
Y
γ
is a scaling factor and is set
here to 0.6.
⎛
⎞
γ
j
=
N
b
j
=1
CH
ω
X
(
⎝
⎠
TOD
=1
−
b
j
)
− CH
ω
Y
(
b
j
)

(8.22)
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