Flowers and Wedges for Stereology
Luis M. Cruz-Orive
Department of Mathematics, Statistics and Computation, University of Cantabria, Santander, Cantabria, Spain

Recently [1] a new decomposition, inspired in a classical result of O.Varga [7], has been found for the motion invariant density of straight lines in three dimensional Euclidean space, with applications in stereology [1,2]. The new decomposition is called the invariator principle, and it leads to new rotational formulae which express the surface area and the volume of a bounded subset in terms of an observable functional defined in an isotropically oriented section (called a pivotal section) through a fixed point (called the pivotal point). The results have been extended to intrinsic volumes of manifolds in general space forms [4,5].

The purpose of this talk is twofold. First, to present new stereological identities for the surface area and the volume of a nonvoid 'particle', namely a compact nonvoid three dimensional subset with piecewise smooth boundary, and second, to facilitate the computation of the corresponding estimators when the particle is convex by means of a polyhedral approximation, so that a pivotal section is a.s.a convex polygon. The latter purpose pertains to computational geometry. Two cases arise whether the pivotal point is either interior [3] or exterior to the particle (in preparation). The first case is suitable to estimate first order mean individual particle characteristics, and it was compared with classical, alternative estimators for the same material studied in [6].

The second case is relevant to the estimation of second order properties, such as the K-function for surface area and volume defined on a stationary random process of convex particles.


[1] Cruz-Orive, L.M. (2005) J. Microsc. 219, 18-28.

[2] Cruz-Orive L.M. et al. (2010) J.Microsc. 240, 94-110.

[3] Cruz-Orive L.M. (2010) J.Microsc., to appear.

[4] Gual-Arnau, X. & Cruz-Orive L.M. (2009) Diff.Geom. Appl. 27, 124-128.

[5] Gual-Arnau X. et al. (2010) Adv.Appl. Math. 44, 298-308.

[6] Karlsson, L.M. & Cruz-Orive, L.M. (1997) J.Microsc. 186, 121-132.

[7] Varga, O. (1935) Math.Zeitschrift 40, 387-405.

Keywords: Stereology; Invariator principle; Surface area; Volume

Biography: Present position: Professor of Statistics, University of Cantabria in Santander (Spain) since 1994. Interests: Stereology, geometric sampling, stochastic geometry, quantitative microscopy.

Previous positions: Research fellow (University of Madrid, 1969-70), Postdoctoral fellow (University of Sheffield, UK, 1973-76), Research Assistant (University of Bern, CH, 1977-82), Extraordinarius Professor (University of Bern, CH, 1983-94).

Education: Agricultural Engineer (Polytechnic University of Madrid, 1968), Diploma in Statistics (University of Edinburgh, UK, 1971), PhD (Department of Probability and Statistics, University of Sheffield, UK, 1973), Habilitation (University of Bern, CH, 1983).