Image Processing Reference

In-Depth Information

Another distinctive element, the zero element, means that a source yielding this

value has complete determination over the result of the fusion. Such elements also

exist for t-norms and t-conorms.

The property of increasingness is usually imposed on operators and matches what

our intuition tells us.

Boundary conditions, which define the behavior of the operators when the infor-

mation to combine has extreme values, guarantee compatibility with the binary case,

where all of the propositions are simply either true or false (this corresponds to the

constraint of complying with deductive logic imposed by Cox for defining an induc-

tive logic [COX 46]).

The continuity property satisfied by most operators guarantees the robustness of

the fusion. However, this property is not always necessary, since natural phenomena

(particularly time phenomena) are not always continuous.

Idempotence means that providing information that is already available will not

change the fusion result. This property is not systematically imposed. It is verified by

means, the t-norm min and the t-conorm max (and those are the only ones). We might

want, on the contrary, to have the combination of two identical values reinforce or

weaken the overall result. Let us consider the example of identical simultaneous testi-

monies. If the witnesses are plotting together, it is not surprising to see them saying the

same thing and the associated degrees of confidence will therefore be combined in an

idempotent way. Whereas if they are independent, the credibility of what they are say-

ing will be reinforced if they are trusted, or weakened if they are not. Let us note that

the combination rules modeling these behaviors have been known since Bernoulli.

Generally speaking, it is considered that if sources are dependent (in the cognitive

sense), idempotence can be imposed, whereas if they are independent, reinforcement

effects can be needed.

Along the same lines, the nilpotence property will be imposed, for example, to

combine consecutive testimonies, in order to model the deterioration of information

along a chain of witnesses that are not completely reliable. For example, for certain

t-conorms, satisfying this property will help achieve a result equal to 1 by combining

a certain number of measures, which are not all equal to zero. This type of behavior

may be useful when the information is the result of a long processing chain.

The excluded middle and non-contradiction properties, satisfied only for certain

operators, have an accepted interpretation in reasoning terms, in the field of artificial

intelligence and fuzzy reasoning. There are examples in image processing where the

excluded middle is not advisable, whenever there is a need for introducing absence of

knowledge regarding an event and its complements and therefore to relax the compre-

hensiveness constraint applied, for example, in probabilities.

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