Cantor, Fuzzy, Near, and Rough Sets in Image Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 2

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.
1.2 Cantor Set
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).
1.3 Near Sets
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,