Random sets theory and its applications to stereology

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Abstract
SUMMARY: In order to study objects forming a sub‐set A of euclidean space, mathematical morphology uses structuring figures B and notes the frequency of events such as ‘B hits A’, ‘B is included into A’ etc. Thus, a probabilistic formalism is associated with this experimental technique and facilitates its interpretation. If A is considered as a closed set, we obtain a random closed‐sets theory, closely connected with integral geometry. The functionals T defined by T(K) = P(A ∩ K ≠ Ø) for K compact are characterized as alternating capacities of infinite order. Interesting classes of functionals T are obtained if A is indefinitely divisible or semi‐markovian. At last, the mathematical notion of granulometry (size distribution) is studied by using an axiomatic method.

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