Image Processing Reference

In-Depth Information

12.3.1

Description-based Set Intersection Nearness Measure

Because we want to consider the problem of comparing objects in pairs of disjoint sets such

as separate images and, yet, we want to gather together from the disjoint sets those objects

that have feature-values in common, we introduce a description-based set intersection. This

problem is of interest because its solution provides a formal basis for an image resemblance

measure considered in the context of tolerance near sets. To solve this problem, we introduce

a description-based intersection of sets.

Definition 5 Description-Based Tolerance Class Intersection (Peters and Henry, 2009).

LethO,

X

i

Y

F

ibe a perceptual system and let X, Y⊆O,B⊆

F

and let

C

,

C

j
⊆X
/∼
=
B

, Y
/∼
=
B
,

respectively. The notation

C

i

denotes x
i/
∼
=
B
, a tolerance class for x
i
∈X and

C

j

denotes

y
j /
∼
=
B
, a tolerance class for y
j
∈Y . Then

i
\

∼

j

i
×

j

C

=
B
C

={(x, y)∈

C

C

:kφ(x)−φ(y)k
2
≤ε}.

TABLE 12.3

Feature Values for a Pair of Samples

X

φ

Y

φ

x
1
0.7

y
1
0.8

x

0.75

y

0.85

2

2

Example 1 Sample Tolerance Class Intersections

ε = 0.2,

(12.2)

X
/
∼
=
φ,0.2

={{x
1
, x
2

}},

(12.3)

Y
/
∼
=
φ,0.2

={{y

, y

}},

(12.4)

1

2

\

X

1

Y

1

C

=
B,0.2
C

={x

, y

},

(12.5)

1

1

∼

\

X

1

Y

2

C

=
B,0.2
C

={x

, y

},

(12.6)

1

2

∼

\

X

2

Y

1

C

=
B,0.2
C

={x

, y

},

(12.7)

2

1

∼

\

X

2

Y

2

C

=
B,0.2
C

={x

, y

}.

(12.8)

2

2

∼

Notice, then, that this example illustrates the fact that the magnitude of
C

i
T
∼

j

=
B
C

can

equal the product of the sample sizes, i.e.,

i
\

∼

=
C

i
·
C

j
.

X

Y

j

X

Y

C

=
B,0.2
C

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