3D Distance Distributions - Where are the Points?

In cytometric measurements of fluorescence color marked molecular elements (e.g. genes in the nucleus) by microscopic imaging, in standard cases 2D projections of the whole object or an appropriate section are obtained. In addition, the object has usually been distorted by biochemical treatments. Therefore, the problem of reconstructing the object or computing the postition of the marked region leads to several mathematical questions. A convenient measure in the microscopic images are distances of two points. Interestingly, for many types of given point densities (e.g. by splines) on the ball, the resulting (projected) distance distributions can be derived analytically. The principle question, namely the inverse problem of determining the point densities (e.g. the localisation of chromosomal domains in the nucleus) from the statistical distribution of the (projected) distances, is far from being solved.