Image Processing Reference

In-Depth Information

slowly forming on window ledges?

-Fragment of a Philosophical Poem.

-Z. Pawlak & J.F. Peters, 2002.

The basic idea in the near set approach to object recognition

is to compare object descriptions. Sets of objects
X
,
Y

are considered near each other if the sets contain objects

with at least partial matching descriptions.

-Near sets. General theory about nearness of objects,

-J.F. Peters, 2007.

TABLE 1 . 1

Nomenclature

Symbol

Interpretation

O

,

X

,

Y

Set of perceptual objects,
X

,

Y

⊆

O

,

A

⊂

X

,

x

∈

X

,

y

∈

Y
,

F
,B

Sets of probe functions,

B ⊆
F
,
φ
i
∈ B

,

i
th
probe function representing feature of
x
,

φ
i
(
x
)

φ
i
:
X

→
ℜ
,

φ
B

(
x
)

(

φ
1
(
x
)

,
φ
2
(
x
)

,...,
φ
i
(
x
)

,...,
φ
k
(
x
)),description of
x
of length
k
,

ε

ε
∈
ℜ

(reals) such that

ε
≥

0,

·
i
)
2
)
2

i
=1
(

·
2

= (

,

L
2

(Euclidean) norm,

∑

=
B,
ε

{

(
x

,

y
)

∈

O

×

O
:

φ

(
x
)

−
φ

(
y
)

2
≤
ε
}

, tolerance relation,

=
B

shorthand for =
B,
ε
,

⊂
=
B,
ε
∀

x
=
B,
ε

is a preclass in =
B,
ε
),

A

x

,

y

∈

A

,

y
(
i.e.
,
A
=
B,
ε

tolerance class, maximal preclass of =
B,
ε
,

C
=
B,
ε

X
=
B,
ε
Y
.

X

B,
ε

Y

X
resembles (is near)
Y

⇐⇒

Set Theory Law 1 Near Sets

Near sets contain elements with similar descriptions.

Near sets are disjoint sets that resemble each other (Henry and Peters, 2010). Resemblance

between disjoint sets occurs whenever there are observable similarities between the objects in the

sets. Similarity is determined by comparing lists of object feature values. Each list of feature values

defines an object's description. Comparison of object descriptions provides a basis for determining

the extent that disjoint sets resemble each other. Objects that are perceived as similar based on

their descriptions are grouped together. These groups of similar objects can provide information

and reveal patterns about objects of interest in the disjoint sets. For example, collections of digital

images viewed as disjoint sets of points provide a rich hunting ground for near sets. For example,

near sets can be found in the favite pentagona coral fragment in Fig. 1.1a from coral reef near Japan.

If we consider the greyscale level, the sets
X

Y
in Fig. 1.1b are near sets, since there are many pixels

in
X
with grey levels that are very similar to pixels in
Y
.

,

Near sets are a generalization of rough sets. It has been shown that every rough set is, in fact, a

near set but not every near set is a rough set Peters (2007b). Near set theory originated from an

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