The problem:
A two dimensional distribution of circular objects allows to determine the
threedimensional distribution. In mathematics this is callled often the tomatoe salat
problem (Tomatoes of different sizes are cut into slices, when only having the slices,
what is the size distribution of the original tomatoes ?)
One has to keep in mind that (as always in mathematical execat proofs) the
propositions are ideal. I.e. only in case of an infinite set of slices the reconstruction
is exact.
The idea how it works:
The size distribution of diameters of slices of a
monodispersive ball distribution (all balls have the same size) when cut at arbitrary
planes can be calculated and depends only of the density of balls.
When analyzing the slice cut of a non monodispersive ball distribution it is clear
that the largest diameters in the cut stems from the fact that the largest balls is cut in
its middle. One can now subtract the contribution of the largest balls from the total set.
The resulting set again has largest diameters. They contribute from cuts from the next
smaller class of balls cut in the middle. Again the contribution of the next smaller class
can be subtracted from the actual distribution.
And so on.
For a precise mathematical treatment one can consult the following articles:
Guenther Bach, "Kugelgrößenverteilung und Verteilung
der Schnittkreise", Institut für Mathematik B,
Technische Hochschule, Braunschweig.
Guenther Bach, "Size Distribution of Particles derived
from the size of their sections", Technische
Hochschule, Braunschweig.
Real situations:
To treat real situations one needs some help to brigde the
difference to mathematical idealization:
One can add cut planes to increase statistic precision, one can treat the
probematical small radii and one can jude the precision of the result by comparing the
volume density to the area density (which can be evaluated exactly).