Rough Fuzzy Measures in Image Segmentation and Analysis - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 212

measure. These measures include summing up fuzzy membership values of all objects from
the given set, in our case objects of (generalized) lower and upper approximations. Other
choice present different types of fuzzy membership functions.
Probabilistic and Non-Probabilistic Entropy
During calculation of rough entropy value (Pal et al., 2005), many type of entropy are
possible to be taken into account, for example Shannon entropy, Renyi entropy, Tsallis
entropy. All these entropies are generally probabilistic entropies, meaning that probability
of all possible states equals to unity.
Fuzzy Rough Entropy
= Fuzzy Rough Entropy− e
2

·roughness(l)·log(roughness(l));
In the presented solution, this condition has not been considered and roughness vales
are not equalized to probability distribution. In case of probabilistic distribution, given
n possible states, entropy attains maximal value in situation with all states having equal
probability of 1/n. In rough entropy framework, roughness value for each cluster is contained
in the interval [0, 1]. In such manner, total rough entropy depends on summing up all partial
rough entropies. Total rough entropy attains the maximum in case of all partial entropies
equal its maximum. Maximum value for partial (cluster) entropy

−R·log(R)

This partial cluster entropy reaches the maximum in case of roughness value equal to 1/e.
Rough entropy framework searches for optimal clustering solution that boundary region is
approximately equal 1/3.

11.4.3 Crisp-Crisp Distance RECA

Standard Crisp - Crisp Distance RECA algorithm as proposed in (Malyszko and Stepaniuk,
2008) incorporates computation of lower and upper approximations for the given cluster
centers and considering these two set cardinalities during calculation of roughness and
further rough entropy clustering measure. Rough Entropy Clustering Algorithm flow has
been presented in Algorithm 1. Rough measure general calculation routine has been given
in Algorithm 2. In all presented algorithms, before calculations, lower and upper cluster
approximations should be set to zero. For each data point x i , distance to the closest cluster
C l is denoted as d min

dist = d(x i , C l ) and approximations are increased by value 1 of clusters
C m that satisfy the condition:

|d(x i , C m )−d min

dist |≤ dist

11.4.4 Fuzzy-Crisp Difference RECA

In Fuzzy-Crisp Difference RECA setting, for the given point, lower or lower and upper
approximation values are incremented not arbitrary by 1, but are increased by this point
cluster membership value. In this way, fuzzy concept of belongings to overlapped classes
has been incorporated. Taking into account fuzzy membership values during lower and
upper approximation calculation, should more precisely handle imprecise information that
imagery data consists of.
Fuzzy membership value µ C l (x i )∈[0, 1] for the data point x i ∈U in cluster C l (equiva-
lently X l ) is given as