Image Processing Reference

In-Depth Information

Notice (O,
∼
=
B
) is an example of a tolerance space. The relation
∼
=
B
in Defn. 3 defines a

covering of O instead of partitioning O because an object can belong to more than one class.

Let A⊂
∼
=
B
. A is a preclass in
∼
=
B
, if∀x, y∈A, x
∼
=
B
y, i.e. kφ
B
(x)−φ
B
(y) k
2
≤

ε (Schroeder and Wright, 1992b). Of course, the idea of a preclass can be generalized to any

metric space. A maximal preclass of tolerance
∼
=
B,ε
is called a class. The particular form

of distance-based tolerance relation
∼
=
B,ε
used in this chapter works well for image analysis,

applied mathematics, engineering and biomedicine, where
∼
=
B,ε
is used to express inaccuracy

of measurement or what Poincare calls an indistinguishable difference in the intensity of a

sensation (Poincare, 1913). In other words, a set A⊂
∼
=
B
is a tolerance class if, and only if

A is a maximal preclass.

The following simple example highlights the need for a tolerance relation as well as demon-

strates the construction of tolerance classes from sample data. Let X
/
∼

=
B,ε
denote a covering

of X defined by
∼
=
B,ε
. Consider the 20 objects in Table 12.2 where|φ(x
i
)|= 1. Put ε = 0.1

and obtain the following tolerance classes:

X
/
∼

=
B
= {{x

, x

, x

, x

},{x

, x

, x

, x

, x

},

1

8

10

11

1

9

10

11

14

{x

, x

, x

, x

},

2

7

18

19

{x

, x

, x

},

3

12

17

{x

},

{x
5
, x
6
, x
15
, x
16
},{x
5
, x
6
, x
15
, x
20
},

{x
6
, x
13
, x
20
}}.

, x

, x

},{x

, x

4

13

20

4

18

Observe that each pair of objects x, y in a tolerance class satisfies the conditionkφ(x)−

φ(y)k
2
≤ε, and many of the objects appear in more than one class.

TABLE 12.2

Tolerance Class Example

x
i

φ(x)

x
i

φ(x)

x
i

φ(x)

x
i

φ(x)

x
1
.4518

x
6
.6943

x
11
.4002

x
16
.6079

x

.9166

x

.9246

x

.1910

x

.1869

2

7

12

17

x
3
.1398

x
8
.3537

x
13
.7476

x
18
.8489

x

.7972

x

.4722

x

.4990

x

.9170

4

9

14

19

x
5
.6281

x
10
.4523

x
15
.6289

x
20
.7143

Definition 4 Tolerance Near Sets (Peters, 2009)

LethO,

be

a set of probe functions representing image features. LetBcontains probe functions used

to measure features of subimages in X, Y⊂O. A set X is perceptually near a Y within

the perceptual systemhO,

F

ibe a perceptual system, where O is a set of images (sets of points) andB⊆

F

F

i(i.e., (X ./
F,ε

Y )) iff there are x∈X and y∈Y and there is

such that x
∼
=
B,ε
y.

B⊆

F

12.3

Resemblance Measures

Tolerance classes can be viewed as structural elements in representing an image. The moti-

vation for using tolerance classes in perceptual image analysis is the conjecture that visual

perception in the human perception is performed in the class level rather than pixel level.

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