Image Processing Reference

In-Depth Information

In Figures 8.8 to 8.10, examples of ordered tolerance matrix is shown for more

images. The images are from the Berkeley segmentation dataset (Martin, Fowlkes,

Tal, and Malik, 2001).

Each image size is 321

×

481 which is divided into 160

×

subimages of size 30

30. The only probe function used is the average gray level

values between 0 and 255. The value of

in equation 8.8 is equal to 20 where the

gray values have not been normalized.

8.2.3

Nearness Relations and Near Sets

Nearness relations were introduced in the context of a perceptual system

O,
F

by James Peters (Peters and Wasilewski, 2009) after the introduction of near set

theory in 2007 (see (Peters, 2007c),(Peters, 2007b) and (Peters and Wasilewski,

2009)). These relations are defined between
sets of perceptual objects
. Therefore, a

nearness relation

R

is a subset of

P

(

O

)

×P

(

O

).

DEFINITION 8.8
Weakly Nearness Relation

Let
O,
F
be a perceptual system and let
X, Y ⊆ O
.

Aset

X

is weakly near to a set

Y

within a perceptual system

O,
F

and is shown

X
F
Y

with

, if and only if the following condition is satisfied:

∃ x ∈ X, y ∈ Y, B⊆
F

x ∼
B
y

Consequently, nearness relation is defined on

such that

P

(

O

) as follows:

F
=

{

(

X, Y

)

∈P

(

O

)

×P

(

O

)

| X
F
Y }

(8.14)

Example 8.6

Figure 8.7 shows two images and their corresponding subimages (25

×

25 pixels

each).

Let

X

and

Y

denotes the set of all subimages in image 1 and image 2

respectively. Let

O,
F

be a perceptual system and let
B⊆
F
and
B
=
{φ
1
(
x
)
}
, where
φ
1
(
x
)=
gray
(
x
)is

the gray scale value of subimage

O

=

X ∪ Y

be the set of all subimages in two images. Let

x

.

X

Y

X
F
Y

Images

and

are then weakly near to each other (

) because they have

elements

x ∈ X

and

y ∈ Y

with matching descriptions (

x ∼
B
y

).

Nearness Relation

DEFINITION 8.9

Let

O,
F

be a perceptual system and let

X, Y ⊆ O

.Aset

X

is near to a set

Y

within a perceptual system

O,
F

and is shown with

X
F
Y

, if and only if the

following condition is satisfied (Peters and Wasilewski, 2009):

∃ x ∈ X, y ∈ Y, A,B⊆
F
,f ∈
F

and also
∃ A ∈ O
/∼
A
,B∈ O
/∼
B
,C∈ O
/∼
C
such that
A, B ⊆ C

Consequently, the nearness relation is defined on
P
(
O
) as follows:

F
=

{

(

X, Y

)

∈P

(

O

)

×P

(

O

)

| X
F
Y }

(8.15)

Search WWH ::

Custom Search