Image Thresholding using Generalized Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 41

exactly definable sets RX and RX in U , which are obtained as

= [ {Y∈U/R : Y⊆X} RX

(3.1)

= [ {Y∈U/R : Y∩X 6=∅} (3.2) RX

In the above, RX and RX are respectively called the R-lower approximation and the R-
upper approximation of X. In essence, the pair of sets < RX, RX > is the representation
of any arbitrary set X⊆U in the approximation space < U, R >, where X can not be
defined. As given in (Pawlak, 1991), an inexactness measure of the set X can be defined as

ρ R (X) = 1− |RX|
|RX|

(3.3)

where|RX|and|RX|are respectively the cardinalities of the sets RX and RX in U . The
inexactness measure ρ R (X) is called the R-roughness measure of X and it takes a value in
the interval [0, 1].

3.2.2 The Lower and Upper Approximations of a Set
The expressions for the lower and upper approximations of the set X depends on the type
of relation R and whether X is a crisp (Klir and Yuan, 2005) or a fuzzy (Klir and Yuan,
2005) set. Here we shall consider the upper and lower approximations of the set X when
R denotes an equivalence, a fuzzy equivalence, a tolerance or a fuzzy tolerance relation and
X is a crisp or a fuzzy set.
When X is a crisp or a fuzzy set and the relation R is a crisp or a fuzzy equivalence
relation, we consider the expressions for the lower and the upper approximations of the set
X as

RX = {(u, M (u))|u∈U}

RX = {(u, M (u))|u∈U} (3.4)

where

= X
Y∈U/R

M (u) m Y (u)×inf

ϕ∈U

max(1−m Y (ϕ), µ X (ϕ))

= X
Y∈U/R

M (u) m Y (u)×sup

ϕ∈U

min(m Y (ϕ), µ X (ϕ))

(3.5)

where the membership function m Y represents the belongingness of every element (u) in
the universe (U ) to a granule Y∈U/R and it takes values in the interval [0, 1] such that

P Y m Y (u) = 1, and µ X , which takes values in the interval [0, 1], is the membership function
associated with X. When X is a crisp set, µ X would take values only from the set{0, 1}.
Similarly, when R is a crisp equivalence relation m Y would take values only from the set

{0, 1}. In the above, the symbols P (sum) and×(product) respectively represent specific
fuzzy union and intersection operations (Klir and Yuan, 2005), which are chosen judging
their suitability with respect to the underlying application of measuring ambiguity.
In the above, we have considered the indiscernibility relation R⊆U×U to be an
equivalence relation, that is, R satisfies crisp or fuzzy reflexivity, symmetry and transitivity
properties (Klir and Yuan, 2005). We shall also consider here the case when the transitivity
property is not satisfied. Such a relation R is said to be a tolerance relation (Skowron