Image Processing Reference

In-Depth Information

DEFINITION 8.6
Tolerance Preclass

Aset
A ⊂
=
B,ε
is a preclass in
=
B,ε
if, and only if
∀x, y ∈ A, x
=
B,ε
y
.

DEFINITION 8.7
Tolerance Class

Aset
A ⊂
=
B,ε
is a tolerance class in =
B,ε
if, and only if
A
is a maximal preclass.

where
=
B,ε
is defined in (8.8) with respect to a given set of probe functions

B

.Let

x
/
=
B,ε

x

. Tolerance classes can overlap and

hence the set of all tolerance classes is a
covering
of

denote a maximal preclass containing

O

denoted by

O
/
=
B,ε
. Recall

that a cover is a family of subsets of

O

whose union is

O

and their intersection is

not necessarily empty.

O
/
=
B,ε
=

{x
/
=
B,ε
| x ∈ O}

(8.9)

x∈O
x
/
=
B,ε
=

O

(8.10)

Example 8.3

Figure 8.3b shows the image in example 8.2 and its subimages. Let

O,
F

be a

perceptual system where

O

denotes the set of 25

×

25 subimages. The image is

divided into 100 subimages of size 25

of

all the 100 subimages. Let
B
=
{φ
1
(
x
)
}⊆
F
where
φ
1
(
x
)=
gray
(
x
) is the average

gray scale value of pixels in subimage

×

25 and can be shown as a set

X

=

O

1. A

subimage
x
has been selected and the marked subimages in the figure belong to a

tolerance class that is represented by the selected subimage because their gray level

values are close to the gray value of subimage

x

, normalized between 0 and 1. Let

=

.

x

within the tolerance level

.

Example 8.4

The simple image in example 8.1
(Figure 8.1)
is considered here again. For each

given subimage, the corresponding tolerance class has been obtained by finding all

the subimages that have the average gray scale values within the tolerance range

(

1) of the average gray value of the given subimage. Figure 8.4 shows all the

tolerance classes which are calculated for

=

.

∀x ∈ O

after removing the redundant

classes. Note that

x
1
/
=
B,ε

=

x
7
/
=
B,ε

=

x
8
/
=
B,ε
,

x
2
/
=
B,ε

=

x
3
/
=
B,ε

=

x
5
/
=
B,ε

and

x

9
/
=
B,ε
and hence there are 4 tolerance classes in total. The set of all

tolerance classes is a covering of
O
and is shown with
O
/
=
B

=

x

6
/
=
B,ε

=
{x
/
=
B
| x ∈ O}
=

{x

4
/
=
B,
,x

5
/
=
B,
,x

8
/
=
B,
,x

9
/
=
B,
, }

.

Tolerance Matrix

In order to demonstrate a tolerance space and all its tolerance classes, a
tolerance

matrix
is defined here to show the tolerance relation between pairs of perceptual

objects. Each row in a tolerance matrix represents one tolerance class and each

column represents one perceptual object (subimage). Corresponding to each probe

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