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A design‐based proof of Wicksell's integral equation
Author(s) -
Jensen E. B.
Publication year - 1984
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.1365-2818.1984.tb00545.x
Subject(s) - randomness , planar , aggregate (composite) , realization (probability) , mathematics , distribution (mathematics) , integral equation , mathematical analysis , position (finance) , cross section (physics) , physics , computer science , quantum mechanics , statistics , materials science , computer graphics (images) , finance , economics , composite material
SUMMARY Wicksell's integral equation (1925) describes, for an aggregate of spherical particles, the relation between the distribution of sphere diameters and the distribution of diameters of circular profiles observed on a planar section. Usually, this integral equation is proved under the assumption that the particle aggregate is a realization of a stationary stochastic process while the position of the planar probe is allowed to be arbitrary. In this note, it is shown that, under the alternative assumption that diameters of circular profiles are sampled by means of a planar FUR ( f ixed orientation u niform r andom) probe, Wicksell's equation holds for a deterministic aggregate of spherical particles with arbitrary positions and sizes. Since the proof is based on the randomness generated by the sampling (the design), the proof may be characterized as design‐based. It can thus be concluded that Wicksell's equation may be applied to any aggregate of spherical particles if diameters are sampled by means of random (FUR) planar probes.