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 suggest that it may replace, for various
purposes, 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 - the
reconstruction of the inchoate distribution from its proximity
function - 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, apart from reflections.
While there is a finite algorithm, it is readily seen that 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.