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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