Image Processing Reference

In-Depth Information

operators. We will present the main ones, then give a few indications on how to choose

a fusion operator according to its properties and its behavior.

An important feature, common to every theory, of these combination operators is

that they provide us with a result of the same nature as the functions we started with

(the closure property) and therefore with the same interpretation in terms of impreci-

sion and uncertainty. Therefore, they make it possible not to make any partial binary

decision before the combination takes place, which could lead to inconsistencies that

would be difficult to eliminate. The decision is only made at the very end, based on

the result of the combination.

In fuzzy set and possibility theory, a number of combination modes are possi-

ble [DUB 85, YAG 91]. Among the major operators, we can mention in particular

t-norms, t-conorms [MEN 42, SCH 83], means [GRA 95, YAG 88], symmetric sums

and operators that take into account conflict or source reliability measures [DEV 93,

DUB 92a], as we saw in section 8.5. From here on, the letters
x
,
y
, etc. refer to the

values we wish to combine, i.e. values in [0
,
1] that therefore represent
μ
i

or
π
j
(
C
i
)

in this case.

The choice of a fusion operator is made according to several criteria presented in

[BLO 96b].

A first criterion is the operator's behavior. Strict, lenient or cautious behaviors are

expressed in mathematical form as conjunction, disjunction, or compromise. Let
x
and

y
be two real numbers (in [0
,
1]) representing the degrees of confidence to combine.

The combination of
x
and
y
by an operator
F
is described as:

- conjunctive if
F
(
x, y
)

≤

min(
x, y
) (corresponding to a strict behavior);

- disjunctive if
F
(
x, y
)

≥

max(
x, y
) (lenient behavior);

- compromise if
x

≤

F
(
x, y
)

≤

y
if
x

≤

y
and
y

≤

F
(
x, y
)

≤

x
otherwise

(cautious behavior).

This distinction is not sufficient to categorize operators whose behaviors are not

always the same. This is why the classification defined in [BLO 96b] describes oper-

ators not only as conjunctive and disjunctive, but also depending on their behavior

according to the values of the information to combine. The three classes suggested

correspond to:

- context independent constant behavior (CICB) operators: the result depends only

on the values to combine (the calculation involves no other information) and the

behavior is the same regardless of what those values are;

- context independent variable behavior (CIVB) operators: the behavior depends

on the numerical values of the information to fuse;

- context dependent (CD) operators, for example, of more comprehensive knowl-

edge such as the reliability of the sensors, or the conflict between the sources.

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