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Random sets theory and its applications to stereology
Author(s) -
Matheron G.
Publication year - 1972
Publication title -
journal of microscopy
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.569
H-Index - 111
eISSN - 1365-2818
pISSN - 0022-2720
DOI - 10.1111/j.1365-2818.1972.tb03708.x
Subject(s) - euclidean geometry , formalism (music) , mathematics , euclidean space , probabilistic logic , structuring , axiomatic system , pure mathematics , axiom , set (abstract data type) , interpretation (philosophy) , discrete mathematics , computer science , geometry , statistics , art , musical , finance , programming language , economics , visual arts
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.