Image Processing Reference
InDepth Information
The proposed method is compared with
tolerance nearness measure
(tNM, a pre
viously developed method by C. Henry and J. Peters (Hassanien, Abraham, Peters,
Schaefer, and Henry, 2009; Peters and Wasilewski, 2009; Henry and Peters, 2009b)
and also with a simple histogram based method based on comparing the cumulative
histograms of gray level values. The results of comparison is presented in example
8.7.
Tolerance nearness measure: tNM
Tolerance nearness measure (tNM) is based on the idea that if one considers the
union of two images as the set of perceptual objects, tolerance classes should contain
almost equal number of subimages from each image.
tNM between two images
(Henry and Peters, 2009b) is defined as follows:
Suppose
X
and
Y
are the sets of perceptual objects (subimages) in image 1 and
image 2. Let
z
/
=
B,ε
denote a maximal preclass containing
z
.
Z
=
X ∪ Y
is the set
of all perceptual objects in the union of images and for each
z ∈ Z
the tolerance
class is shown as:
z
/
=
B,ε
=
{s ∈ Z
 φ
B
(
z
)
− φ
B
(
s
)
≤
(8.23)
The part of the tolerance class that is a subset of
X
is named as [
z
/
=
B,ε
]
⊆X
and
similarly, part of the tolerance class that is a subset of
Y
is named [
z
/
=
B,ε
]
⊆Y
.
Therefore:
[
z
/
=
B,ε
]
⊆X
{x ∈ z
/
=
B,ε
 x ∈ X}⊆z
/
=
B,ε
(8.24)
[
z
/
=
B,ε
]
⊆Y
{y ∈ z
/
=
B,ε
 y ∈ Y }⊆z
/
=
B,ε
(8.25)
z
/
=
B,ε
=[
z
/
=
B,ε
]
⊆X
∩
[
z
/
=
B,ε
]
⊆Y
(8.26)
Subsequently, a tolerance nearness measure is defined as the weighted average of
the closeness between the cardinality (size) of sets [
z
/
=
B,ε
]
⊆X
and the cardinality of
[
z
/
=
B,ε
]
⊆Y
where the cardinality of
z
/
=
B,ε
is used as the weighting factor.
min(

[
z
/
=
B,ε
]
⊆X
 , 
[
z
/
=
B,ε
]
⊆Y

)
1
tNM
×
)
×z
/
=
B,ε

=
(8.27)
max(

[
z
/
=
B,ε
]
⊆X
 , 
[
z
/
=
B,ε
]
⊆Y

z
/
=
B,ε

z
/
=
B,ε
z
/
=
B,ε
Histogram similarity measure: HSM
For the sake of comparison, a third similarity measure is also defined here to compare
distributions (histograms) of gray scale values in images. Therefore a
histogram
similarity measure
is defined here as the absolute difference between CDF of the
gray scale values in two images similar to equation 8.22
⎛
⎞
γ
j
=
N
b
j
=1
CH
g
X
(
⎝
⎠
HSM
=1
−
b
j
)
− CH
g
Y
(
b
j
)

(8.28)
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