The reconstruction problem in microdosimetry

A generalised formulation of microdosimetry clarifies the linkages between the spatial distribution of energy deposits, their proximity function, and the specific energy. The role of the proximity function suggests that it may replace the inchoate distribution. The Fourier transform of the proximity function is the product of the Fourier transform of the inchoate distribution with its conjugate. This operation causes the loss of phase information, and the reconstruction problem can, therefore, not be resolved by a mere deconvolution. For any finite point pattern one can, however, show that its proximity function permits, in principle, the reconstruction. Numerical examples with 2-dimensional patterns of up to 30 points have consistently led to unique solutions. While there is a finite algorithm, the number of steps becomes excessive when the number of points in the pattern increases. The reconstruction problem can, thus, be solved in principle but not necessarily in practice. A more general approach must thus be based on numerical optimisation. The algorithm starts with an assumed initial point pattern and utilises a suitable measure for the difference between its proximity function and that of the original pattern. Minimising this difference can lead to the original or to a similar pattern. With simple algorithms one obtains convergence only for patterns of few points; but improved optimisation methods are likely to provide more general solutions. (author)