Image Processing Reference
In-Depth Information
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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