• Luis Manuel Cruz-Orive University of Cantabria (E-Santander)




flower area, invariator, nucleator, stereology, surface area, surfactor, volume, wedge volume


It is shown that, for a three dimensional particle  (namely an arbitrary compact domain with piecewise smooth boundary in R^3) the mean wedge volume defined on a given pivotal section is equal to the average nucleator estimator of the particle volume defined on that section. Further, if the particle is convex and it contains the pivotal point, then the flower area of a given pivotal section equals the average surfactor estimator defined on that section. These results are intended to throw some light on the standing conjecture that the functional defined on a pivotal section according to the invariator has a unique general expression. As a plus, the former result leads to a computational formula for the mean wedge volume of a convex polygon which is much simpler than the one published recently, and it is valid whether the fixed pivotal point is interior or exterior to the particle.


Cruz-Orive LM (1987) Particle number can be estimated using a disector

of unknown thickness: the selector. J Microsc 145: 121--142.

Cruz--Orive LM (2005) A new stereological principle for test lines in

D. J Microsc 219: 18--28.

Cruz-Orive LM (2008) Comparative precision of the pivotal estimators of

particle size. Image Anal Stereol 27: 17--22.

Cruz-Orive LM (2011) Flowers and wedges for the stereology of

particles. J Microsc 243: 86--102.

Cruz-Orive LM, Ramos-Herrera ML and Artacho-Perula E (2010) Stereology

of isolated objects with the invariator. J Microsc 24: 94--110.

Gundersen HJG (1988) The nucleator. J Microsc 151: 3-21.

Gual-Arnau X, Cruz-Orive LM and Nuno-Ballesteros JJ (2010) A new

rotational integral formula for intrinsic volumes in space forms. Adv

ApplMath 44: 298--308.

Hansen LV, Nyengaard JR, Andersen JB and Jensen EBV (2011) The

semi-automatic nucleator. J Microsc 242: 206--215.

Jensen EBV (1991) Recent developments in the stereological analysis of

particles. Ann Inst Statist Math 43: 455--468.

Jensen EB and Gundersen HJG (1987) Stereological estimation of surface

area of arbitrary particles. Acta Stereol 6/ Suppl III: 25--30.

Jensen EB and Gundersen HJG (1989) Fundamental stereological formulae

based on isotropically orientated probes through fixed points with

applications to particle analysis. J Microsc 153: 249--267.

Jensen EBV and Rataj J (2008) A rotational integral formula for


volumes. Adv Appl Math 41: 530--560.

Karlsson, LM and Cruz-Orive, LM (1997) Estimation of mean particle size

from single sections. J Microsc 186, 121--132.




How to Cite

Cruz-Orive, L. M. (2012). UNIQUENESS PROPERTIES OF THE INVARIATOR, LEADING TO SIMPLE COMPUTATIONS. Image Analysis and Stereology, 31(2), 89–98. https://doi.org/10.5566/ias.v31.p89-98



Original Research Paper