Image Processing Reference

In-Depth Information

and Crisp z
i

and Crisp z
i

(a) Crisp Υ
T

(b) Fuzzy Υ
T

and Fuzzy z
i

and Fuzzy z
i

(c) Crisp Υ
T

(d) Fuzzy Υ
T

(e) Crisp Υ
T

and

(f) Fuzzy Υ
T

and

S
ω

: G×G→{0, 1}

S
ω

: G×G→{0, 1}

(g) Crisp Υ
T

and

(h) Fuzzy Υ
T

and

S
ω

:

G

×

G

→[0

,

1]

S
ω

:

G

×

G

→[0

,

1]

FIGURE 3.4: The different forms that the lower and upper approximation of Υ
T
can take when

used to get the grayness ambiguity measure

The grayness ambiguity measure Λ of I is obtained as a function of T , which characterizes

the underlying set Υ
T
, as follows

%
ω
(Υ
T
) log
β
%
ω
(Υ
T
)

β

+ %
ω
(Υ
T
) log
β
%
ω
(Υ
T
)

β

Λ
ω
(T ) =−
1

2

(3.30)

Note that, the above expression is obtained by using %
ω
(Υ
T
) and %
ω
(Υ
T
) in the proposed

logarithmic (L) class of entropy functions given in (3.8), instead of roughness measures.

When the proposed exponential (E) class of entropy functions is used, we get

%
ω
(Υ
T
)β
1−%
ω
(Υ
T
)
+ %
ω
(Υ
T
)β
1−%
ω
(Υ

T
)

1

2

{

Λ
ω
(T ) =

(3.31)

It should be noted that the values %
ω
(Υ
T
) and %
ω
(Υ
T
) in (3.29) are obtained by considering

'weighted cardinality' measures instead of cardinality measures, which are used for calcu-

Search WWH ::

Custom Search