Image Processing Reference

In-Depth Information

of the segmentation process; additionally, fuzzy logic allows to represent the knowledge

about the given problem in terms of linguistic rules with meaningful variables, which is the

most natural way to express and interpret information. Fuzzy image processing (Aboul Ella

et al., 2004; Kerre and Nachtegael, 2000; Nachtegael, Van-Der-Weken, Van-De-Ville, Kerre,

Philips, and Lemahieu, 2001; Sandeep and Rene, 2006; Sushmita and Sankar, 2005; Rosen-

feld, 1983) is the collection of all approaches that understand, represent and process the

images, their segments and features as fuzzy sets. An image I of size M xN and L gray

levels can be considered as an array of fuzzy singletons, each having a value of membership

denoting its degree of brightness relative to some brightness levels.

5.2.2 Rough sets

Due to space limitations we provide only a brief explanation of the basic framework of rough

set theory, along with some of the key definitions.

A more comprehensive review can be

found in sources such as (Polkowski, 2002).

Rough sets theory provides a novel approach to knowledge description and to approxi-

mation of sets. Rough theory was introduced by Pawlak during the early 1980s (Pawlak,

1982) and is based on an approximation space-based approach to classifying sets of objects.

In rough sets theory, feature values of sample objects are collected in what are known as

information tables. Rows of a such a table correspond to objects and columns correspond

to object features.

LetO,Fdenote a set of sample objects and a set of functions representing object features,

respectively. Assume that B⊆F, x∈O. Further, let x
∼
B

denote

x
/
∼
B

={y∈O|∀φ∈B, φ(x) = φ(y)},

i.e., x∼
B
y (description of x matches the description of y). Rough sets theory defines three

regions based on the equivalent classes induced by the feature values: lower approximation

BX, upper approximation BX and boundary BN D
B
(X). A lower approximation of a set X

contains all equivalence classes x
/
∼
B
that are proper subsets of X, and upper approximation

BX contains all equivalence classes x
/
∼
B
that have objects in common with X, while the

boundary BN D
B
(X) is the set BX\BX, i.e., the set of all objects in BX that are not

contained in BX. Any set X with a non-empty boundary is roughly known relative, i.e.,

X is an example of a rough set.

The indiscernibility relation∼
B
(also written as Ind
B
) is a mainstay of rough set theory.

Informally,∼
B
is a set of all classes of objects that have matching descriptions. Based on

the selection of B (i.e., set of functions representing object features),∼
B
is an equivalence

relation that partitions a set of objectsOinto classes (also called elementary sets (Pawlak,

1982)). The set of all classes in a partition is denoted byO
/
∼
B
(also byO/Ind
B
). The set

O/Ind
B
is called the quotient set. A
nities between objects of interest in the set X⊆O

and classes in a partition can be discovered by identifying those classes that have objects

in common with X. Approximation of the set X begins by determining which elementary

sets x
/
∼
B
∈O
/
∼
B

are subsets of X.

5.2.3 Neural networks

Neural networks (NN) is an Artificial Intelligent (AI) methodology based on the composition

of the human brain, as well as made up of a wide network of interconnecting processors.

The basic parts of every NN are the processing elements, connections, weights, transfer

functions, as well as the learning and feedback laws.

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