# Stereology Course

14 – 15 December, 2011, in Koll. G (1532-214), Aarhus University

## Lecturer

Luis M.Cruz-Orive

## Keywords

Antithetic isector, computational geometry, Crofton formulae, estimation variance, flower set, geometrical probability, invariator, isotropic Cavalieri, motion invariance, particle, pivotal tessellation, rat brain, stereology, surface area, test line, test plane, test system, uniqueness properties, volume, wedge set, weighted sampling.

## Tentative programme

### Wed 14th December 2011. 12:00 - 15:00

1. Classical construction of motion invariant test probes. Crofton formulae. Applications to stereology.

• Exercise 1: "Estimating the volume and the external surface area of a rat brain with the isotropic Cavalieri design."

2. Motion invariant test lines in $~\mathbb{R}^3$. Invariator construction. Applications to stereology.

• Exercise 2: "Estimating the volume and the external surface area of a rat brain with the invariator. Empirical assessment of the error variance."

3. Case of a convex particle. Surface area in terms of the flower set.

### Thu 15th December 2011. 12:00 - 15:00

1. Computation of the flower area and the wedge volume for an interior pivotal point.
1. The pivotal section is convex with smooth boundary of known parametric
coordinates.
2. The pivotal section is a convex polygon.
2. Uniqueness properties of the invariator. Connections with the nucleator and the surfactor. Computational implications.
3. Open questions, final discussion.

## References

Prior to the seminars, participants may want to have a (very) cursory look at the following papers.

[1] Cruz-Orive, L.M. (2001/02) Stereology: meeting point of integral
geometry, probability, and statistics. In memory of Professor Luis A.Santalo (1911-2001). Special issue (Homenaje a Santalo) ofMathematicae Notae 41, 49-98.

[2] Cruz--Orive LM (2005) A new stereological principle for test lines in 3D. J Microsc 219: 18-28.

[3] Cruz-Orive LM, Ramos-Herrera ML and Artacho-Perula E (2010)Stereology of isolated objects with the invariator. J Microsc 24: 94-110.

[4] Cruz-Orive LM (2011) Flowers and wedges for the stereology of particles. J Microsc 243: 86-102.

[5] Cruz-Orive LM (2011) Uniqueness properties of the invariator, leading to simple computations. Internal Report 1/2011. MATESCO, Universidad de Cantabria. Submitted for publication.

# Lecture Notes

Lecture notes part 2