Rough Fuzzy Measures in Image Segmentation and Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 211

Now, let R represent an arbitrary relation on U. With respect to R, it is possible to define
the left and right neighborhoods of an element x in U in the following way

I Lef t
R

(x) ={y|yRx}, and,

I Right
R

(x) ={y|xRy}.

In this convention, if R is symmetric, then I Lef t
R

(x) = I Right
R

(x). The left (or right) neighbor-

hood I Lef t

R (x) (or I Righ R (x)) becomes an equivalence class containing x if R is an equivalence
relation. If R is a tolerance relation, then we obtain I lef t
R

= I Right
R

and the approximation

space (U, (I Lef t

R , I Righ R ), v) is reduced to (U, I R , v). For an arbitrary relation R, by replacing
equivalence class with the right neighborhood, the operators U P P and LOW from P (U )
to itself are defined by

LOW (AS R , X) ={x|I Right (x)⊆X}, and

R

U P P (AS R , X) ={x|I Right (x)∩X 6= φ},

R

analogously to definitions from the previous subsection, LOW (AS R , X) is called a lower
approximation of X and U P P (AS R , X) an upper approximation of X. The pair

(LOW (AS R , X), U P P (AS R , X))

is referred to as a rough set based on R. The set LOW (AS R , X) consists of those elements
whose right neighborhoods are contained in X, and the set U P P (AS R , X) consists of those
elements whose right neighborhoods have a non-empty intersection with X. Clearly, in case
of R being an equivalence relation, then these definitions are equivalent to the original rough
set definitions.
In the context of proposed rough entropy framework, objects from the universe U, are
assigned to the clusters represented by clusters centers and with each cluster are defined
lower and upper cluster approximations. In contrast to established solutions based on an
equivalence relation, a new binary relation R has been introduced that describes, for a given
point x, clusters or cluster centers related to x. Object x is in relation R with cluster center
center C.
Generalized Lower and Upper Approximations
The binary relation that has been presented induces generalized lower and upper ap-
proximations according to the definitions given in this subsection. The lower generalized
approximation contains objects that are uniquely located in the nearest neighborhood (ac-
cording to predefined criteria) to this cluster center and only to this cluster center. The
upper generalized approximation contains objects that are close to this cluster center but
additionally may belong to one or more other clusters. Generalized lower and upper ap-
proximation for the given cluster will not be the same, but such a situation is possible and
represents crisp clustering, with objects assigned uniquely to one cluster.
Similarity Measures
Similarity measures are based on distance threshold, fuzzy distance threshold and fuzzy
threshold measures.
Roughness
Most often, the roughness value R(AS B , X) is defined numerically in the way described in
the previous subsection. In the rough entropy clustering setting, however, this is not the only
alternative.

During roughness calculation other measures are possible besides a counting