Image Processing Reference

In-Depth Information

1. identify a sample space
O
and a set

F

to formulate a perceptual system

O

,
F

, and then

2. introduce a tolerance relation

τ
ε

that defines a cover on
O
.

,
τ
ε
), where
X
is the visual field of the right

Zeeman (Zeeman, 1962) introduces a tolerance space (
X

∈

eye and

ε

is the least angular distance so that all points indistinguishable from
x

X
are within

ε

of

x
. In this case, there is an implicit perceptual system

O

,
F

, where
O
:=
X
consists of points that

are sources of reflected light in the visual field and

F

contains probes used to extract feature values

from each
x

∈

O
.

1.3.5 Sets of Similar Images

Consider

O

,
F

, where
O
consists of points representing image pixels and

F

contains probes used

. Then introduce tolerance relation
=
B,
ε

to extract feature values from each
x

∈

O
.Let

B ⊆
F

to

define a covers on
X

,

Y

⊂

O
. Then, in the case where
X

,

Y
resemble each other,
i.e.
,
X

B,
ε

Y
,

then measure the degree of similarity (nearness) of
X

Y
(a publicly available toolset that makes it

possible to complete this example for any set of digital images is available at (Henry and Peters,

2010, 2009a)). See
Table 1.1
(also, (Peters and Wasilewski, 2009; Peters, 2009b, 2010)) for details

about the bowtie notation

,

B,
ε

B,
ε

used to denote resemblance between
X
and
Y
,
i.e.
,
X

Y

(1.3a) Lena

(1.3b) Lena TNS

FIGURE 1.3: Lena Tolerance Near Sets (TNS)

1.3.6 Tolerance Near Sets

In near set theory, the trivial case is excluded. That is, an element
x

X
is not considered near itself.

In addition, the empty set is excluded from near sets, since the empty set is never something that we

perceive,
i.e.
, a set of perceived objects is never empty. In the case where one set
X
is near another

set
Y
, this leads to the realization that there is a third set containing pairs of elements
x

∈

Y

with similar descriptions. The key to an understanding of near sets is the notion of a description.

The description of each perceived object is specified a vector of feature values and each feature is

,

y

∈

X

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