Image Processing Reference
In-Depth 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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