Fuzzy Mathematical Morphology - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 198

From Equation 9.18, a wide class of operators can be defined with different
values of λ. These are

n

1. λ n () ( x =− ≥ 1 x ) , n 1 (9.21)

This is equivalent to Zadeh's concentration operation on linguistic hedges:

1
11 1

− x

nx n

2. λ n x () =

(( )) ,

≥ 1 (9.22)

+ / −

This is equivalent to Sugeno's λ complementation:

1 x ln()

ln()

3
2

158 3. λ n () x = − , n ≥ = . (9.23)

n

1 + x 2

n

1 x

e − 1 ln( ln(( − e 12

12

+ ) / e ))

4. λ n () x = , n ≥ = 1 396 . (9.24)

e − 1 ln( / )

Equations 9.21 and 9.22 are widely used. With different selections of λ( x ) and
with different values of ' n ', different eroded and dilated images are obtained.

9.3.3 Fuzzy Morphology Using t -Norms and t -Conorms
by De Baets and Kerre and Bloch and Maitre

DAx 5 () sup( = i By xAy ⎣ − ), () ⎦

yS ∈

(9.25)

( )

EAx 5 () inf = u cByxAy ⎣ ⋅ ( − ), () ⎦

yS ∈

where
i is any t -norm (fuzzy intersection)
u is any t -conorm (fuzzy union)
c is a complement
A is an image
B is a structuring element

Using different t -norms ( T ) and t -conorms ( T *), different forms of erosion
and dilation can be generated:

1. Zadeh t -operators

* max( ,)

T = x y

T = min( ,) x y