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Felix-Klein-Kolloquium | Vortrag »Wicksell’s Corpuscle Problem«
28. Oktober 2014, 17:15 - 18:30
Motivated by the question of how the size distribution of biological cells can be estimated from two-dimensional microscopy images, the statistician S.D. Wicksell formulated and solved in the 1920ies the »corpuscle problem« for spherical particles: Assume that we observe an appropriately randomized planar section of a collection of spherical particles in three-dimensional space. Can the distribution of the sphere radii be estimated from the distribution of the section circle radii?
In the first part of my talk I will give an overview of results on the corpuscle problem and its variants since its first appearance in Wicksell’s works. We will see how this problem is related to solving an Abel integral equation and mention extensions to more general particle shapes, such as (rotationally symmetric) ellipsoids.
In the past, Wicksell’s problem has always been considered in a setting, where the section plane is chosen independently of the particles. In the last part of the talk we consider Wicksell’s problem in a formulation that is more adapted to modern confocal microscopy, where each particle is sampled individually by an isotropic random plane through a reference point – think of the nucleus of a cell. We will consider the spherical case and show that measurements in such section profiles allow for a determination of the cell radius distribution.
The last part of this talk on the corpuscle problem in local stereology is based on joint work with Ólöf Thórisdóttir.