• Markus Kiderlen Aarhus University, Aarhus
  • Karl-Anton Dorph-Petersen Aarhus University, Aarhus



Cavalieri estimator, perturbed spacing, randomized Newton-Cotes quadrature, variance approximations


The Cavalieri method allows to estimate the volume of a compact object from area measurements in equidistant parallel planar sections. However, the spacing and thickness of sections can be quite irregular in applications. Recent publications have thus focused on the effect of random variability in section spacing, showing that the classical Cavalieri estimator is still unbiased when the stack of parallel planes is stationary, but that the existing variance approximations must be adjusted. The present paper considers the special situation, where the distances between consecutive section planes can be measured and thus where Cavalieri’s estimator can be replaced by a quadrature rule with randomized sampling points. We show that, under mild conditions, the trapezoid rule and Simpson’s rule lead to unbiased volume estimators and give simulation results that indicate that a considerable variance reduction compared to the generalized Cavalieri estimator can be achieved.

Author Biographies

Markus Kiderlen, Aarhus University, Aarhus

Centre for Stochastic Geometry and Advanced Bioimaging,  Department of Mathematics

Karl-Anton Dorph-Petersen, Aarhus University, Aarhus

Centre for Stochastic Geometry and Advanced Bioimaging 

Further affiliations: 2. Translational Neuropsychiatry Unit, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark.
 3. Translational Neuroscience Program, Department of Psychiatry, University of Pittsburgh, Pittsburgh, PA, USA.


bibitem[Baddeley etal(2006)]{B_DP_J06}

Baddeley, A., Dorph-Petersen, K.-A. and Jensen, E.B.V. (2006):

A note on the stereological implications of irregular spacing of sections.

emph{J.~Microsc.} {bf 222}, 177--81.

bibitem[Baddeley & Jensen(2005)]{BadJen05} Baddeley, A. and Jensen, E.B.V. (2005):

emph{Stereology for Statisticians.} Chapman & Hall, New York.

bibitem[Heveling & Last(2005)]{HevelingLast05} Heveling, M. and Last, G. (2005):

Characterization of Palm measures via bijective point shifts. emph{Ann.~Probab.} {bf 33},


bibitem[Mecke(1975)]{Mecke75} Mecke, J. (1975): Invarianzeigenschaften allgemeiner Palmscher Mass{}e. emph{Math. Nachr.} {bf 65}, 335-–44.

bibitem[Schneider & Weil(2008)]{SchneiderWeil} Schneider, R. and Weil, W.~(2008): emph{Stochastic and Integral Geometry.} Springer, New York.

bibitem[Selmer(1958)]{Selmer58} Selmer, E.S. (1958): Numerical integration by non-equidistant ordinates. emph{Nordisk Matematisk Tidskrift} {bf 6}, 97--108.

bibitem[Stoer & Bulirsch(1980)]{StBu80} Stoer, J. and Bulirsch, R. (1980): emph{Introduction to Numerical Analysis.} Springer, New York.

bibitem[Ziegel etal(2010)]{Z_B_DP_J10}

Ziegel, J., Baddeley, A. Dorph-Petersen, K.-A. and Jensen, E.B.V.

(2010): Systematic sampling with errors in sample locations.

emph{Biometrika} {bf 97}, 1--13.

bibitem[Ziegel etal(2011)]{Z_J_DP_11}

Ziegel, J., Jensen, E.B.V. and Dorph-Petersen, {K.-A.} (2011):

Variance estimation for generalized Cavalieri estimators.

emph{Biometrika} {bf 98}, 187--98.




How to Cite

Kiderlen, M., & Dorph-Petersen, K.-A. (2017). THE CAVALIERI ESTIMATOR WITH UNEQUAL SECTION SPACING REVISITED. Image Analysis and Stereology, 36(2), 133–139.



Technical Note