Image Enhancement - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 90

3. Entropy by Burillo and Bustince

1 1

N − M −

1

∑ ∑

() (

1 − π ( g ))

EA MN ( ) = π ge

Aij

3 IFS A ij

×

j = 0 i = 0

Intuitionistic fuzzy entropy ( IFE ) is calculated from any of the entropies for
all the λ values. The optimum value of λ that corresponds to the maximum
value of the entropy values is written as

λ = max( IFEA ( ; λ ))

opt IFS

So, in the IF domain, the image is represented as

{ }

A = g , (; μλνλ 01 g ), (; g ) g ∈ ,, , … L − 1

IFSopt _ A opt A opt

Atanassov's operator is applied to A IFS _ opt to deconstruct an IF image to a
fuzzy image. With different values of α, different images are obtained in the
fuzzy domain. Atanassov's operator is written as [3]

( ) =

{ }

∈ [

]

DA xx x , μ () + απν () () ,† x +− ( 1 απ ) () xx X ∈ , α 0 1 ,

α IFSopt _ A A A A

The maximum index of fuzziness intuitionistic defuzzification [20] is used to
select the optimum value of α. In computing the maximum index of defuzzi-
fication, the linear index of fuzziness is required.
The linear index of fuzziness of fuzzy set A is

n

1

∑

γ () x = min( ( , μ x 1 − μ ( x ))

l A i A i

2 X

i = 1

where | X | is the cardinality of X , n = | X |. Substituting min t -norm with the
product operator, the modified index of fuzziness is written as

n

1

∑

(5.6)

γ () x = μ ( ( x 1 − μ ( x ))

i A i A i

2 X

i = 1

To find α opt , the maximization index of fuzziness is required:

{ }

(5.7)

α = max( ( γ DA ))

opt α IFS opt _

α ∈ [,] 01