Image Processing Reference
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and ω
Y
(y) for the set Y
/∼
B,ε
. Now, let F
ω
X
(ω) and F
ω
Y
(ω) be the empirical commutative
distribution functions (CDF) of the functions ω
X
(x) and ω
Y
(y) respectively, when x∈X
and y∈Y .
The Tolerance Overlap Distribution (TOD) nearness measure is defined as
follow:
T OD = 1−
Z
ω=1
ω=0
(F
ω
X
(ω)−F
ω
Y
(ω))dω
γ
.
(12.14)
12.3.3
Tolerance Class SizeBased Nearness Measure (Henry and Pe
ters, 2008)
A tolerance class sizebased nearness (tNM) measure 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. Therefore, the tolerance nearness
measure between two digital images is calculated in the following way.
Suppose X and Y are the sets of perceptual objects (subimages) in image 1 and image 2.
Z = X∪Y is the set of all perceptual objects in the union of images and for each z∈Z,
the tolerance class of an element z of Z (Bartol et al., 2004) (denoted z
/∼
B,ε
in this chapter)
for our application is defined in (12.15).
z
/∼
B,ε
={s∈Z  kφ
B
(z)−φ
B
(s)k≤ε},
(12.15)
i.e.,
={s∈Z  (z, s)∈
∼
=
B,ε
}.
z
/∼
B,ε
The part of the tolerance class z
/∼
B,ε
that is a subset of X is denoted [z
/∼
B,ε
]
⊆X
and,
similarly, part of the tolerance class z
/∼
B,ε
that is a subset of Y
is denoted [z
/∼
B,ε
]
⊆Y
.
Therefore:
[z
/∼
B,ε
]
⊆X
,{x∈z
/∼
B,ε
x∈X}⊆z
/∼
B,ε
,
(12.16)
[z
/∼
B,ε
]
⊆Y
,{y∈z
/∼
B,ε
y∈Y}⊆z
/∼
B,ε
,
(12.17)
z
/∼
B,ε
= [z
/∼
B,ε
]
⊆X
∩[z
/∼
B,ε
]
⊆Y
.
(12.18)
Subsequently, the measure tNM 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 in (12.19).
×
X
z
/∼
B,ε
min([z
/∼
B,ε
]
⊆X
, [z
/∼
B,ε
]
⊆Y
)
max([z
/∼
B,ε
]
⊆X
, [z
/∼
B,ε
]
⊆Y
)
×z
/∼
B,ε
.
1
tN M =
X
(12.19)
z
/∼
B,ε

z
/∼
B,ε
For a more detailed explanation of this measure considered within the general framework of
near sets, see (Henry and Peters, 2009b).
12.3.4
Hausdorff Distance and Image Correspondence Measure
The Hausdorff distance is used to measure the distance between sets in a metric space (Haus
dorff, 1914) (see (Hausdorff, 1962) for English translation), and is defined as
d
H
(X, Y ) = max{sup
x∈X
inf
y∈Y
d(x, y), sup
y∈Y
inf
x∈X
d(x, y)},
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