Image Processing Reference

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O
/∼
B
=

{x
/∼
B
| x ∈ O},

(8.4)

x∈O
x
/∼
B
=

O,

(8.5)

∀x, y ∈ O

(

x
/∼
B
)

∩

(

y
/∼
B
)=

∅.

(8.6)

Example 8.2

Figure 8.3a shows an image of size 256

×

256 pixels and their subimages. Let

O,
F

be a perceptual system where

O

denotes the set of 25

×

25 subimages.

A) The original image

B) A subimage in the covering

A) The original image

B) A subimage in the covering

(b)

(a)

(a)

(b)

C) The equivalence class on a white background

D) The equivalence class marked on image

C) The equivalence class on a white background

D) The equivalence class marked on image

(c)

(d)

(c)

(d)

(8.3a) An equivalent class

(8.3b) A tolerance class

FIGURE 8.3: An image and one of its equivalence (Left) and tolerance (Right)

classes

Let
B
=
{φ
1
(
x
)
}⊆
F
is the set of only one probe function

φ
1
where

φ
1
(
x
)=

gray

. The two marked subimages

are perceptually indiscernible with respect to their gray level values and hence they

are named an
equivalence class
. The equivalence class is shown both individually

(c) and on top of the image where the rest of image is blurred (d).

(

x

) is the average gray scale value of subimage

x

Perceptual indiscernibility relation can also be defined in a weak sense as follows

Perceptual Weak Indiscernibility Relation

(Peters and

DEFINITION 8.3

Wasilewski, 2009; Peters, 2009)

Let

O,
F

be a perceptual system. Let

B

=

{φ
1
,φ
2
, ..., φ
l
}

and

x, y ∈ O

. A percep-

tual weak indiscernibility relation

∼
B
is defined relative to

B

as follows

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