Image Processing Reference

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(1.4a) Photographer

(1.4b) Photographer TNS

FIGURE 1.4: Photographer Tolerance Near Sets

represented by what is known as a probe function that maps an object to a real value. Since our

main interest is in detecting similarities between seemingly quite disjoint sets such as subimages in

an image or pairs of classes in coverings on a pair of images, a near set is defined in context of a

tolerance space.

Definition 2 Tolerance Near Sets
(Peters, 2010)

Let

O

,
F

be a perceptual system.

Put

ε
∈
ℜ
,B ⊂
F

.Let
X

,

Y

⊂

O
denote disjoint sets with

coverings determined by a tolerance relation =
B,
ε
. Sets
X

,

Y
are tolerance near sets if, and only if

there are preclasses
A

⊂

X

,

B

⊂

Y
such that
A

B,
ε

B
.

1.3.7 Near Sets in Image Analysis

The subimages in Fig. 1.3b and Fig. 1.4b delineate tolerance classes (each with its own grey level)

subregions of the original images in Fig. 1.3a and Fig. 1.4a. The tolerance classes in these images

are dominated by (light grey), (medium grey) and (dark grey) subimages along with a few

(very dark) subimages in Fig. 1.3b and many very dark subimages in Fig. 1.4b. From Def. 2,

it can be observed that the images in Fig. 1.3a and Fig. 1.4a are examples of tolerance near sets,

i.e.
,
Image
Fig
.
1
.
4
a
F
,
ε

Image
Fig
.
1
.
3
a
). Examples of the near set approach to image analysis can

be found in,
e.g.
, (Henry and Peters, 2007, 475-482, 2008, 1-6, 2009a; Gupta and Patnaik, 2008;

Peters, 2009a,b, 2010; Peters and Wasilewski, 2009; Peters and Puzio, 2009; Hassanien, Abraham,

Peters, Schaefer, and Henry, 2009; Meghdadi, Peters, and Ramanna, 2009; Fashandi, Peters, and

Ramanna, 2009) and in a number of chapters of this topic.

From set composition Law 1, near sets are Cantor sets containing one or more pairs of objects

(
e.g.
, image patches, one from each digital image) that resemble each other as enunciated in Def. 2,

i.e.
,
X

,

T

⊂

O
are near sets if, and only if
X

F
,
ε

Y
).

1.4 Fuzzy Sets

A fuzzy set is a class of objects with a continuum

of grades of membership.

-Fuzzy sets, Information and Control 8

-L.A. Zadeh, 1965.

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