Image Thresholding using Generalized Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 48

approximation of fuzzy sets (that is, when the set X considered in the previous section
is fuzzy) can be used to quantify grayness ambiguity due to both fuzzy boundaries and
rough resemblance. Whereas, the entropy measures based on the generalization of rough
set theory regarding the approximation of crisp sets (that is, when the set X considered in
the previous section is crisp) can be used to quantify grayness ambiguity only due to rough
resemblance.
Now, we shall obtain the grayness ambiguity measure by considering the fuzzy boundaries
of regions formed based on global gray value distribution and the rough resemblance between
nearby gray levels. The image is considered as an array of gray values and the measure
of consequence of the incompleteness of knowledge about the universe of gray levels in
the array quantifies the grayness ambiguity. Note that, the measure of incompleteness of
knowledge about a universe with respect to the definability of a set must be used here, as
the set would be employed to capture the vagueness in region boundaries.
Let G be the universe of gray levels and Υ T be a set in G, that is Υ T ⊆G, whose
elements hold a particular property to extents given by a membership function µ T defined
on G. Let O I be the graylevel histogram of the image I under consideration. The fuzzy
boundaries and rough resemblance in I causing the grayness ambiguity are related to the
incompleteness of knowledge about G, which can be quantified using the proposed classes
of entropy measures in 3.2.3.
We shall consider Υ T such that it represents the category 'dark areas' in the image I and
the associated property 'darkness' given by the membership function µ T shall be modeled
using the Z-function (Klir and Yuan, 2005) as given below

8
<

1 l≤T−∆

1−2 "

# 2 T−∆≤l≤T

(l− (T− ∆))

2∆

µ T (l) = Z(l; T, ∆) =

2 "

# 2 T≤l≤T + ∆ ; l∈G (3.24)

:

(l− (T+∆))

2∆

0 l≥T + ∆

where T and ∆ are respectively called the crossover point and the bandwidth. We shall
consider the value of ∆ as a constant and that different definitions of the property 'darkness'
can be obtained by changing the value of T , where T∈G.
In order to quantify the grayness ambiguity in the image I using the proposed classes of
entropy measures, we consider the following sets

Υ T = {(l, µ T (l))|l∈G}

Υ T

= {(l, 1−µ T (l))|l∈G} (3.25)

The fuzzy sets Υ T and Υ T considered above capture the fuzzy boundary aspect of the
grayness ambiguity. Furthermore, we consider limited discernibility among the elements
in G that results in vague definitions of the fuzzy sets Υ T and Υ T , and hence the rough
resemblance aspect of the grayness ambiguity is also captured.
Granules, with crisp or fuzzy boundaries, are induced in G as its elements are drawn
together due to the presence of limited discernibility (or indiscernibility relation) among
them and this process is referred to as the graylevel granulation. We assume that the indis-
cernibility relation is uniform in G and hence the granules formed have a constant support
cardinality (size), say, ω. Now, using (3.4), (3.5) and (3.6), we get general expressions
for the different lower and upper approximations of Υ T and Υ T

obtained considering the