Image Processing Reference

In-Depth Information

has a degree of truth based on a t-conorm
T
:

μ
A
∨
B
(
x, y
)=
T
μ
A
(
x
)
,μ
B
(
y
)
,

and a negation has a degree of truth defined by a fuzzy complementation
c
:

¬
A
(
x
)=
c
μ
A
(
x
)
.

μ

S

In the case of variables with values in a product space, i.e.
X
with values in

and

V

Y
with values in

, conjunction is interpreted as a cartesian product. The degree of

truth of:

X
is
A

and

Y
is
B

is then written:

μ
A
×
B
(
x, y
)=
t
μ
A
(
x
)
,μ
B
(
y
)
.

Now let us consider the implication. In classical logic, we have:

A
=

⇒

(
B
)

⇐⇒

(
B
or non-
A
)
,

[8.85]

and therefore the implication is expressed based on a disjunction and a negation. By

using the same equivalence in the fuzzy case, a fuzzy implication is defined based on a

t-conorm (disjunction) and a complementation (negation). Let
A
and
B
be non-fuzzy

sets. The degree to which
A
implies
B
is defined by:

Imp(
A, B
)=
T
c
(
A
)
,B

[8.86]

where
T
is a t-conorm and
c
is a complementation.

In the case where
A
and
B
are fuzzy, we have:

T
c
μ
A
(
x
)
,μ
B
(
x
)
.

[8.87]

Imp(
A, B
)=inf

x

The following table sums up the major fuzzy implications used in other works for

fuzzy reasoning:

−

T
(
x, y
)=max(
x, y
)

max(1

a, b
)

Kleene-Diene

T
(
x, y
)=min(1
,x
+
y
) min(1
,
1

−

a
+
b
)

Lukasiewicz

T
(
x, y
)=
x
+
y

−

xy

1

−

a
+
ab

Reichenbach

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