Image Processing Reference

In-Depth Information

Examples of probe functions are the colour, size, weight of an object. Probe functions

are used to describe an object to determine the characteristics and perceptual similarity

of perceivable physical objects. Perceptual information is always presented with respect to

probe functions just as our senses define our perception of the world. For example, our

ability to view light in the “visible spectrum” rather than infra red or microwaves spectra

defines our perception of the world just as the selection of probe functions constrains the

amount of perceptual information available for extraction from a set of objects.

Definition 2 Perceptual System (Peters and Wasilewski, 2009).

A perceptual systemhO,

iis a real valued total deterministic information system where O

is a non-empty set of perceptual objects and

F

F

is a countable set of probe functions.

The notion of a perceptual system admits a wide variety of different interpretations that

result from the selection of sample perceptual objects contained in a particular sample space

O. Examples of perceptual objects include observable organism behaviour, growth rates,

soil erosion, microfossils, events containing the outcomes of experiments such as energizing a

network, microscope images, MRI scans, and the results of searches for relevant web pages.

The description of an object x∈O with probe functionsBis given by

φ
B
(x) = (φ
1
(x), φ
2
(x), . . . , φ
i
(x), . . . , φ
l
(x)),

where l is the length of the description vector φ and each φ
i
(x) is a probe function that

describes the object x. Let d
i
= φ(x)−φ(y) denote the i
th
vector difference relative to

descriptions φ(x), φ(y) for objects x, y∈O. Then let d
T
, d denote row and column vectors

of description differences, respectively, i.e.,

2

4

3

5
.

d
1

. . .

d
k

d
T

= (d
1
. . . d
k
), d =

Finally, the overall distance computed using (12.1) is the L
2
normkdk
2

for a vector d, i.e.,

v
u
u
t

X

k

kdk
2
= (d
T
d)
2
=

d
2
i
.

(12.1)

i

=1

In general,k·k
2

denotes the length of a vector in L
2
space (Janich, 1984).

Definition 3 Perceptual Tolerance Relation (Peters, 2009)

LethO,

the tolerance relation
∼
=
B,ε

F

ibe a perceptual system and let ε∈

R

. For everyB⊆

F

is defined as follows:

∼
=
B,ε
={(x, y)∈O×O :kφ
B
(x)−φ
B
(y)k
2
≤ε}.

For notational convenience, this relation can be written
∼
=
B
instead of
∼
=
B,ε
with the under-

standing that
is inherent to the definition of the tolerance relation.

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