Image Processing Reference

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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

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