MaPhySto logo

MaPhySto and StocLab Summer School on

Stereology and Geometric Tomography

Sandbjerg Manor, 20-25 May, 2000

StocLab logo

The aim of the summer school was to give an overview of modern stereology and its relation to geometric tomography, including both the mathematical and statistical theory and the practical applications.


Stereology is the area of stochastics dealing with statistical inference about spatial structures from geometric samples of the structure such as two-dimensional sections and one-dimensional probes. The development of stereological methods involve the use of advanced mathematical tools, especially from geometric measure theory and integral geometry. Stereology is now in world-wide use in many areas of biology and medicine, most importantly in neuroscience and cancer grading. Other areas of application are geology, metallography and mineralogy.

A disector consists of a reference plane and a look-up plane, a distance h apart. Illustration of the sampling relevant for estimation of length in R3, based on projections on a vertical plane.

Geometric tomography is closely related to stereology, as is apparent from its definition: ``geometric tomography is the area of mathematics dealing with the retrieval of information about a geometric object from data about its sections, or projections, or both''. Geometric tomography has connections with convex geometry, geometric probing in robotics, computerized tomography, and other areas. The area is surveyed in the book Geometric Tomography by Richard Gardner.

The summer school was held at Sandbjerg Manor, a conference centre owned by University of Aarhus, situated in the southern part of Jutland, Denmark. The summer school was organized by StocLab (Laboratory for Computational Stochastics) and by MaPhySto (Centre for Mathematical Physics and Stochastics), both Department of Mathematical Sciences, University of Aarhus.

The teaching team consisted of

Lectures by invited researchers in related fields such as convex geometry, stochastic geometry and spatial statistics were given, as well as lectures by the participants of the summer school.

Programme, updated January 19, 2004

(L)   : Lectures
(TE)  : Theoretical Exercises
(SL)  : Special Invited Lectures
(PL)  : Participants' Lectures 
(PE)  : Practical Exercises


Saturday 20 May 2000

15.00-16.00 Coffee

16.00-16.30 Welcome
            Presentation of teachers and participants

16.30-17.10 Eva B. Vedel Jensen:
            Introduction to stereology
            Richard Gardner:
            Introduction to geometric tomography
17.15-18.00 Adrian Baddeley:
            Analogy between stereology and survey sampling (L)
18.00       Dinner

20.30-      Get-together   

Sunday 21 May 2000

09.00-09.45 Richard Gardner:
            Star bodies, chord functions, and section functions (L)
10.00-10.45 Eva B. Vedel Jensen:
            Stereological estimation of number (L)

10.45-11.15 Coffee

11.15-12.00 Niels Væver Hartvig:
            Stereological estimation of number (TE)

12.00-14.00 Lunch

14.00-14.45 Richard Gill:
            Quantum tomography (SL)

14.45-15.15 Coffee

15.15-16.00 Aljosa Volcic:
            Determination of a convex body by sections I (SL)

16.10-16.50 Markus Kiderlen:
            Endomorphisms of convex bodies (PL)

17.00-17.30 Boris Rubin:
            Arithmetrical properties of generalized Minkowski-Funk transforms 
            and small denominators on the sphere (PL)

18.00       Dinner

20.00-21.00 Adrian Baddeley:
            The fractionator (PE)

Monday 22 May 2000

09.00-09.45 Richard Gardner:
            The spherical Radon transform and Funk's theorem (L)

10.00-10.45 Hans Jørgen G. Gundersen:
            Counting and sampling in 3D (demonstration at the microscope)

10.45-11.15 Coffee

11.15-12.00 Adrian Baddeley:
            Stereological sampling designs, volume estimation (L)

12.00-14.00 Lunch
14.00-14.45 Aljosa Volcic:
            Determination of a convex body by sections II

14.45-15.15 Coffee
15.15-16.00 Martin Bøgsted Hansen 
            Statistical aspects of inverse problems (SL)
16.10-16.50 Niels Holm Olsen:
            3D-reconstruction, light microscopy and optics of embryo (PL)
17.00-17.30 Andrew Olenko:
            Closeness of random fields in different multidimensional 
            metrics (PL)

19.00-      Summer school dinner


Tuesday 23 May 2000

09.00-09.45 Richard Gardner:
            Determination by chord and section functions (L)
10.00-10.45 Eva B. Vedel Jensen:
            Length and surface area estimation under isotropy (L)

10.45-11.15 Coffee

11.15-12.00 Kien Kieu:
            Length and surface area estimation under isotropy (TE)
12.00-13.00 Lunch

13.00-17.30 Excursion

18.00       Dinner 


Wednesday 24 May 2000

09.00-09.45 Richard Gardner:
            Affine inequalities and volume estimates (L)
10.00-10.45 Adrian Baddeley:
            Vertical sections (L)

10.45-11.15 Coffee
11.15-12.00 Kien Kieu
            Variance of planar area estimators based on systematic 
            sampling (L)

12.00-14.00 Lunch
14.00-14.45 Boris Rubin:
            Continuous wavelet transforms in geometric tomography (SL)

14.45-15.15 Coffee
15.15-15.45 Eugene Spodarev:
            One isoperimetrical problem for stationary flat processes (PL)

15.45-16.15 Martin Bøgsted Hansen:
            Nonparametric estimation of the chord length distribution (PL)
16.30-17.00 Marta Garcia-Finana:
            Fractional trend of the variance under systematic sampling 
            on R (PL)
17.00-17.30 Stephan Boehm:
            On Laslett's test for Boolean model (PL)

18.00       Dinner

20.00-21.00 Hans Jørgen G. Gundersen:
            How to estimate the volume and surface area of a banana (PE)

Thursday 25 May 2000

09.00-09.45 Eva B. Vedel Jensen:
            Local stereology (L)

10.00-10.45 Kien Kieu:
            Variance of planar area estimators based on systematic 
            sampling (TE)

10.45-11.15 Coffee

11.15-12.00 Hans Jorgen G. Gundersen
            Connectivity (L)

12.00-14.00 Lunch


List of Participants

More Information

For the Summer School a small booklet was made, in which the program, abstracts, small notes, list of participants, etc. were included. You may download the (revised and expanded) booklet.

Please make further inquiries to MaPhySto ( or to the organizer Eva B. Vedel Jensen.

Pictures from the Summer School

This document,, was last modified January 19, 2004. Questions or comments to the contents of this document should be directed to