Image Processing Reference
In-Depth Information
This is obtained using the
IF
generator:
λ
ϕ
() (
x
=− ≥ ∈
1
x
) ,
λ
1
,
x
[,]
01
By varying λ and ω, different representations of images are obtained in the
IF
domain. To obtain the optimum values of ω and λ, the
IF
entropy is used:
{
}
(
)
(
)
1
/
ω
λ
/
ω
ω
ω
1
−
max
1
−
1
−
μ
( )
g
,
1
−
μ
()
g
L
1
−
∑
1
A
A
(5.11)
EA
(;,)
ωλ
=
hg
()
{
}
A
A
(
)
(
)
MN
1
/
ω
λω
/
ω
ω
1
−
min
1
−
1
−
μ
( )
g
,
1
−
μ
()
g
g
=
0
A
A
Entropy is a function of ω and λ. For a constant ω,
E
A
(
A
; ω, λ) attains a max-
imum for a specific value of λ denoted as λ
opt
(ω). The optimum value is
obtained by maximizing the fuzzy entropy. An
IF
image is obtained as
{
}
(
)
=
(
)
∈
A
λωωμ ων λω
( ,
x
, (;),
g
g
;
,
g
012
,,,
…
,
L
−
1
opt
A
A
opt
h
A
(
g
) is the histogram of the fuzzified image.
5.4.3 Entropy-Based Enhancement Method by Chaira (Method III)
The entropy-based enhancement method is suggested by Chaira [6] where
the image is considered fuzzy and so grey levels are imprecise. Due to quan-
tization noise, a grey level
g
in a digital image may be (
g
+ 1) or (
g
− 1). Taking
this into account, for each grey level,
g
, the grey values that are (
g
+ 1) or
(
g
− 1) are replaced by grey level
g
. This is computed for all the grey levels.
The image is initially fuzzified μ
A
(
g
) using Equation 5.1.
From Sugeno's fuzzy complement, the
IF
membership function is given as
=−
−
+⋅
1
μ
λμ
()
()
g
=
+⋅
+⋅
(
1
1
λμ
λμ
) ()
()
g
g
(5.12)
IFS
A
A
μ
()
g
1
A
1
g
A
A
Using Sugeno's fuzzy negation,
1
−
+⋅
x
(5.13)
ϕ
()
x
=
1
λ
x
While computing the non-membership function of an
IF
image, λ in Equation
5.13 is changed (increased) to λ
+ 1. With the change in λ, the non-membership
degree,
ν
IFS
(
, will change (decrease) but will still follow the condition
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