Image Processing Reference

In-Depth Information

tribution of this chapter is an overview of the high utility of fuzzy sets, near sets and rough sets

with the emphasis on how these sets can be used in image analysis, especially in classifying parts

of digital images presented in this topic.

To establish a context for the various sets utilized in this topic, this section briefly presents the notion

of a Cantor set. From the definition of a Cantor set, it is pointed out that fuzzy sets, near sets and

rough sets are special forms of Cantor sets. In addition, this chapter points to links between the three

types of sets that are part of the computational intelligence spectrum. Probe functions in near set

theory provide a link between fuzzy sets and near sets, since every fuzzy membership function is a

particular form of probe function. Probe functions are real-valued functions introduced by M. Pavel

in 1993 as part of a study of image registration and a topology of images (Pavel, 1993). Z. Pawlak

originally thought of a rough set as a new form of fuzzy set (Pawlak, 1981a). It has been shown

that every rough set is a near set (this is Theorem 4.8 in (Peters, 2007b)) but not every near set is a

rough set. For this reason, near sets are considered a generalization of rough sets. The contribution

of this chapter is an overview of the links between fuzzy sets, near sets and rough sets as well as the

relation between these sets and the original notion of a set introduced by Cantor in 1883 (Cantor,

1883).

By a 'manifold' or 'set' I understand any multiplicity,

which can be thought of as one,
i.e.
, any aggregate

[
inbegri f f
] of determinate elements which,

can be united into a whole by some law.

-Foundations of a General Theory of Manifolds,

-G. Cantor, 1883.

...A set is formed by the grouping together

of single objects into a whole.

-Set Theory

-F. Hausdorff, 1914.

In this mature interpretation of the notion of a set, G. Cantor points to a property or law that de-

termines elementhood in a set and “unites [the elements] into a whole” (Cantor, 1883), elaborated

in (Cantor, 1932), and commented on in Lavine (1994). In 1851, Bolzano (Bolzano, 1959) writes

that “an aggregate so conceived that is indifferent to the arrangement of its members I call a
set
”.

At that time, the idea that a set could contain just one element or no elements (null set) was not con-

templated. This is important in the current conception of a near set, since such a set must contain

pairs of perceptual objects with similar descriptions and such a set is never null. That is, a set is a

perceptual near set if, and only if it is never empty and it contains pairs of perceived objects that

have descriptions that are within some tolerance of each other (see Def. 2).

How Near

How near to the bark of a tree are drifting snowflakes,

swirling gently round, down from winter skies?

How near to the ground are icicles,

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