Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 156

distributions that we wish to combine into an overall distribution π . They may repre-
sent, for example, the imprecision on a variable estimated in two different ways, for
which we want an overall estimation.

Let us consider, for example, the case of the conjunctive combination of two pos-
sibility distributions π 1 and π 2 defined on D . This type of combination is well suited
for the case where the distributions overlap at least partially, i.e. when certain classes
are presented as possible by the two sources. If this is not the case, the sources are in
conflict and a possible measure of conflict is:
h π 1 ,π 2 =1 min π 1 ( c ) ,π 2 ( c ) ,

− max

c ∈ D

[8.77]

which represents 1 minus the height of the intersection between the two distributions
(calculated by a min in this equation). The combination can be normalized by this
height, but this would hide the conflict: a possibility of 1 is always assigned to the
classes presented as the most possible by both sources, even if that possibility is low
(this problem is similar to that mentioned in section 7.4 about the conjunctive com-
bination of belief functions). In terms of conflict, the interpretation of this quantity
matches our intuition of triangular or trapezoidal possibility distributions (and more
generally of monomodal possibility distributions), but it is not well suited for forms in
which a single point can generate a strong conflict value, even if the two distributions
are different in that point only.

In the extreme case of completely conflicting distributions, conjunctive combina-
tion leads to an identically zero distributions. A disjunctive combination is then the
preferred method, making it possible to keep all of the data if it is presented as possi-
ble by at least one of the two sources. The underlying hypothesis is that at least one of
the sources is reliable.

Here are a few examples of the possible formulae for π :
π ( s )=max t π 1 ( s ) ,π 2 ( s )

,

h π 1 ,π 2

h π 1 ,π 2

−

, 1 [8.78]

π ( s )=min 1 , t π 1 ( s ) ,π 2 ( s )

,

h π 1 ,π 2

h π 1 ,π 2

+1 − [8.79]

π ( s )= t π 1 ( s ) ,π 2 ( s ) +1 h π 1 ,π 2 ,

− [8.80]

π ( s )=max min π 1 ,π 2
h

.

, min max π 1 ,π 2 , 1 h − [8.81]

The first two forms combine normalized conjunction with a constant distribution
of conflict value, whereas the latter allows us to switch from a strictly conjunctive
combination, when the conflict is equal to zero, to a strictly disjunctive combination,