Stereology with Cylinder Probes

Authors

  • Luis Manuel Cruz-Orive University of Cantabria (E-Santander)
  • Ximo Gual-Arnau University Jaume I. E-12071 Castelló

DOI:

https://doi.org/10.5566/ias.2433

Keywords:

cylinders, integral geometry, motion invariant measures, ratio design, stereology, test system

Abstract

Intersection formulae of Croton type for general geometric probes are well known in integral geometry.  For the special case of  cylinders with non necessarily convex direktrix, however, no equivalent formula seems to exist in the literature. We derive such formula resorting to motion invariant probability elements associated with test systems, instead of using a traditional approach. Because cylinders are seldom used as probes in stereological practice, however, this note is mainly of a theoretical nature.

Author Biographies

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

Dept. of Mathematics, Statistics               
and Computation,  (MATESCO),               
Faculty of Sciences,
University of Cantabria,         
Avda. Los Castros, 48,
E-39005 Santander, Spain

Ximo Gual-Arnau, University Jaume I. E-12071 Castelló

Professor

Department of Mathematics
Institute of New Imaging Technologies

References

Baddeley A, Jensen EBV (2005). Stereology for

statisticians. Boca Raton: Chapman and Hall/ CRC.

Cruz-OriveLM(1980). Best linear unbiased estimators

for stereology. Biometrics 36:595-605.

Cruz-Orive LM (2002). Stereology: meeting point of

integral geometry, probability, and statistics. In

memory of Professor Luis A.Santaló (1911-2001).

Special issue (Homenaje a Santaló), Mathematicae

Notae 41:49-98.

Cruz-OriveLM (2017). Stereology:A historical survey.

Image Anal Stereol 36:156-177.

Cruz-Orive LM, Weibel E (1981). Sampling designs

for stereology. J Microsc 122:235-72.

Hadwiger H (1957) Vorlesungen über Inhalt,

Oberfläche un Isoperimetrie. Berlin: Springer.

HorganGW, Buckland ST, Mackie-Dawson LA (1993).

Estimating three-dimensional line process densities

from tube counts. Biometrics 49:899-906.

Miles RE (1972). Multidimensional perspectives in

stereology. J Microsc 95:181-95.

Rey-Pastor J, Santaló LA (1951). Geometría Integral,

Madrid: Espasa-Calpe.

Santaló LA (1936) Integralgeometrie 5. Über das

kinematische Mass im Raum. Paris: Hermann.

Santaló LA (1976). Integral Geometry and Geometric

Probability. London: Addison-Wesley Publishing

Company Inc.

Schneider R, Weil W (2008). Stochastic and Integral

Geometry, Berlin:Springer.

Voss F, Cruz-Orive LM (2009). Second moment

formulae for geometric sampling with test probes.

Statistics 43:329–65.

Downloads

Published

2020-11-25

How to Cite

Cruz-Orive, L. M., & Gual-Arnau, X. (2020). Stereology with Cylinder Probes. Image Analysis and Stereology, 39(3), 213–218. https://doi.org/10.5566/ias.2433

Issue

Section

Original Research Paper

Most read articles by the same author(s)

1 2 > >>