Image Thresholding using Generalized Rough Sets - Rough Fuzzy Image Analysis: Foundations and Methodologies - page 50

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-