Image Processing Reference

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different indiscernibility relations discussed in Section 3.2.2 as follows

Υ
T
={(l, M

Υ
T
(l))|l∈G}, Υ
T
={(l, M

Υ
T
(l))|l∈G}

Υ
T

={(l, M

(l))|l∈G}, Υ
T

={(l, M

(l))|l∈G}
(3.26)

Υ

{

T

T

Υ

where we have

γ

X

M
Υ
T
(l)

=

m
z
i
(l)×inf

ϕ∈G

max(1−m
z
i
(ϕ), µ
T
(ϕ))

i=1

γ

X

M

Υ
T
(l)

=

m
z
i
(l)×sup

ϕ∈G

min(m
z
i
(ϕ), µ
T
(ϕ))

i=1

γ

X

M

(l)

=

m
z
i
(l)×inf

ϕ∈G

max(1−m
z
i
(ϕ), 1−µ
T
(ϕ))

Υ

{

T

i

=1

γ

X

M

(l)

=

m
z
i
(l)×sup

ϕ∈G

min(m
z
i
(ϕ), 1−µ
T
(ϕ))

(3.27)

T

Υ

i=1

when equivalence indiscernibility relation is considered and we have

M

Υ
T
(l) = inf

ϕ∈G

max(1−S
ω
(l, ϕ), µ
T
(ϕ)), M

Υ
T
(l) = sup

ϕ∈G

min(S
ω
(l, ϕ), µ
T
(ϕ))

M

Υ

(l) = inf

ϕ∈G

max(1−S
ω
(l, ϕ), 1−µ
T
(ϕ)), M

(l) = sup

ϕ∈G

min(S
ω
(l, ϕ), 1−µ
T
(ϕ))
(3.28)

T

{

T

Υ

when tolerance indiscernibility relation is considered. In the above, γ denotes the number

of granules formed in the universe G and m
z
i
(l) gives the membership grade of l in the i
th

granule

i
. These membership grades may be calculated using any concave, symmetric and

normal membership function (with support cardinality ω) such as the one having triangular,

trapezoidal or bell (example, the π function) shape. Note that, the sum of these membership

grades over all the granules must be unity for a particular value of l. In (3.28), S
ω
: G×G→

[0, 1], which can be any concave and symmetric function, gives the relation between any

two gray levels in G. The value of S
ω
(l, ϕ) is zero when the difference between l and ϕ is

greater than ω and S
ω
(l, ϕ) equals unity when l equals ϕ.

The lower and upper approximations of the sets Υ
T
and Υ
T
take different forms depending

on the nature of rough resemblance considered, and whether the need is to capture grayness

ambiguity due to both fuzzy boundaries and rough resemblance or only those due to rough

resemblance. The nature of rough resemblance may be considered such that an equivalence

relation between gray levels induces granules having crisp (crisp

z

i
)

boundaries, or there exists a tolerance relation between between gray levels that may be crisp

(S
ω
: G×G→{0, 1}) or fuzzy (S
ω
: G×G→[0, 1]). When the sets Υ
T
and Υ
T
considered

are fuzzy sets, grayness ambiguity due to both fuzzy boundaries and rough resemblance

would be captured. Whereas, when the sets Υ
T
and Υ
T
considered are crisp sets, only the

grayness ambiguity due to rough resemblance would be captured. The different forms of

the lower and upper approximation of Υ
T
are shown graphically in Figure 3.4.

We shall now quantify the grayness ambiguity in the image I by measuring the conse-

quence of the incompleteness of knowledge about the universe of gray levels G in I. This

measurement is done by calculating the following values

%
ω
(Υ
T
) = 1−
P
l∈G
M
Υ
T
(l)O
I
(l)

i
) or fuzzy (fuzzy

z

z

P
l∈G
M

(l)O
I
(l)

Υ

{

T

Υ
T
(l)O
I
(l)
, %
ω
(Υ
T
) = 1−

P
l∈G
M

P
l∈G
M

(3.29)

(l)O
I
(l)

T

Υ

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