Image Processing Reference
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These two equations are usually not verified by strictly monotonic, archimedean
t-norms and t-conorms.
Any strictly monotonic, archimedean t-norm (or t-conorm) can be defined based
on multiplicative generators, by equivalence with additive generators [CHE 89]:
[0 , 1] 2 ,t ( x, y )= h 1 h ( x ) h ( y ) ,
( x, y )
[8.61]
where h is a strictly increasing function of [0 , 1] into [0 , 1] such that h (0) = 0 and
h (1) = 1. The equivalence with the additive form is obtained simply by defining:
h = e f
[8.62]
where f is an additive generating function.
The most common t-norms and t-conorms in this class are the product and the
algebraic sum:
[0 , 1] 2 ,t ( x, y )= xy,
( x, y )
T ( x, y )= x + y
xy.
[8.63]
The only rational t-norms of this class are the Hamacher t-norms defined by
[HAM 78]:
xy
[0 , 1] 2 ,
( x, y )
xy ) ,
[8.64]
γ +(1
γ )( x + y
where γ is a positive parameter (for γ =1we get the product again). They are illus-
trated in Figure 8.5.
Figure 8.5. Two examples of Hamacher t-norms, for
γ =0
(left)
and
γ =0 . 4
(right)
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