Local Stereology of Tensors of Convex Bodies

  • Eva B. Vedel Jensen
  • Johanna F. Ziegel


In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in ℝ d . Special cases cover a number of well-known local stereological estimators of volume and surface area in ℝ3, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation.


Ellipsoidal approximation Local stereology Minkowski tensors Particle shape Particle orientation Rotational integral geometry 

AMS 2000 Subject Classification

60D05 53C65 52A22 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Auneau-Cognacq J, Ziegel J, Jensen EBV (2013) Rotational integral geometry of tensor valuations. Adv Appl Math 50:429–444CrossRefzbMATHMathSciNetGoogle Scholar
  2. Beisbart C, Dahlke R, Mecke KR, Wagner H (2002) Vector- and tensor-valued descriptors for spatial patterns. In: Morphology of condensed matter, vol 600. Lecture notes in physics. Springer, Berlin, pp 249–271Google Scholar
  3. Beisbart C, Barbosa MS, Wagner H, da F Costa L (2006) Extended morphometric analysis of neuronal cells with Minkowski valuations. Eur Phys J B 52:531–546CrossRefGoogle Scholar
  4. Beneš V, Jiruše M, Slámová M (1997) Unfolding the trivariate size-shape-orientation distribution of spheroidal particles with application. Acta Mater 45(3):1105–1113CrossRefGoogle Scholar
  5. Cruz-Orive LM (1976) Particle size-shape distributions: the general spheroid problem. i. Mathematical model. J Microsc 107:235–253CrossRefGoogle Scholar
  6. Cruz-Orive LM (1978) Particle size-shape distributions: the general spheroid problem. II. Stochastic model and practical guide. J Microsc 112:153–167CrossRefGoogle Scholar
  7. Cruz-Orive LM (2005) A new stereological principle for test lines in three-dimensional space. J Microsc 219:18–28CrossRefMathSciNetGoogle Scholar
  8. Cruz-Orive LM (2008) Comparative precision of the pivotal estimators of particle size. Image Anal Stereol 27:17–22CrossRefzbMATHGoogle Scholar
  9. Cruz-Orive LM (2011) Flowers and wedges for the stereology of particles. J Microsc 243:86–102CrossRefGoogle Scholar
  10. Cruz-Orive LM (2012) Uniqueness properties of the invariator, leading to simple computations. Image Anal Stereol 31:89–98CrossRefMathSciNetGoogle Scholar
  11. Cruz-Orive LM, Hoppeler H, Mathieu O, Weibel ER (1985) Stereological analysis of anisotropic structures using directional statistics. J Roy Statist Soc Ser C 34:14–32zbMATHMathSciNetGoogle Scholar
  12. Denis EB, Barat C, Jeulin D, Ducottet C (2008) 3D complex shape characterizations by statistical analysis: Application to aluminium alloys. Mater Charact 59:338–343CrossRefGoogle Scholar
  13. Dvořák J, Jensen EBV (2012) On semi-automatic estimation of surface area. CSGB Research Report 12-06, Centre for Stochastic Geometry and Advanced Bioimaging, Department of Mathematics, Aarhus University, Denmark. SubmittedGoogle Scholar
  14. Gokhale M (1996) Estimation of bivariate size and orientation distribution of microcracks. Acta Mater 44(2):475–485CrossRefGoogle Scholar
  15. Gundersen HJ (1988) The nucleator. J Microsc 151:3–21CrossRefGoogle Scholar
  16. Hansen LV, Nyengaard JR, Andersen JB, Jensen EBV (2011) The semi-automatic nucleator. J Microsc 242:206–215CrossRefGoogle Scholar
  17. Hug D, Schneider R, Schuster R (2008) Integral geometry of tensor valuations. Adv Appl Math 41:482–509CrossRefzbMATHMathSciNetGoogle Scholar
  18. Jensen EB, Gundersen HJG (1987) Stereological estimation of surface area of arbitrary particles. Acta Stereol 6:25–30Google Scholar
  19. Jensen EBV (1998) Local stereology. World Scientific, LondonzbMATHGoogle Scholar
  20. Jensen EBV, Gundersen HJ (1993) The rotator. J Microsc 170:35–44CrossRefGoogle Scholar
  21. Leopardi P (2006) A partition of the unit sphere into regions of equal area and small diameter. Electron Trans Numer Anal 25:309–327 (electronic)zbMATHMathSciNetGoogle Scholar
  22. Pawlas Z, Nyengaard JR, Jensen EBV (2009) Particle sizes from sectional data. Biometrics 65:216–224CrossRefzbMATHMathSciNetGoogle Scholar
  23. Schröder-Turk GE, Kapfer SC, Breidenbach B, Beisbart C, Mecke K (2011a) Tensorial Minkowski functionals and anisotropy measures for planar patterns. J Microsc 238:57–74CrossRefGoogle Scholar
  24. Schröder-Turk GE, Mickel W, Kapfer SC, Klatt MA, Schaller FM, Hoffmann MJF, Kleppmann N, Armstrong P, Inayat A, Hug D, Reichelsdorfer M, Peukert W, Schwieger W, Mecke K (2011b) Minkowski tensor shape analysis of cellular, granular and porous structures. Adv Mater 23:2535–2553CrossRefGoogle Scholar
  25. Thórisdóttir O, Kiderlen M (2013) The invariator principle in convex geometry (in preparation)Google Scholar
  26. Zähle M (1987) Curvatures and currents for unions of sets with positive reach. Geom Dedicata 23:155–171CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark
  2. 2.Department of Mathematics and Statistics, Institute of Mathematical Statistics and Acturarial ScienceUniversity of BernBernSwitzerland

Personalised recommendations