  Microscopy research often requires recovering particle-size distributions in
three dimensions from only a few (10 - 200) profile measurements in the
section. This problem is especially relevant for petrographic and mineralogical
studies, where parametric assumptions are reasonable and finding distribution
parameters from the microscopic study of small sections is essential. This
paper deals with the specific case where particles are approximately spherical
(i.e. Wicksell's problem). The paper presents a novel approximation of the
probability density of spherical particle profile sizes. This approximation
uses the actual non-smoothness of mineral particles rather than perfect
spheres. The new approximation facilitates the numerically efficient use of the
maximum likelihood method, a generally powerful method that provides the
distribution parameter estimates of the minimal variance in most practical
cases. The variance and bias of the estimates by the maximum likelihood method
were compared numerically for several typical particle-size distributions with
those by alternative parametric methods (method of moments and minimum distance
estimation), and the maximum likelihood estimation was found to be preferable
for both small and large samples. The maximum likelihood method, along with the
suggested approximation, may also be used for selecting a model, for
constructing narrow confidence intervals for distribution parameters using all
the profiles without random sampling and for including the measurements of the
profiles intersected by section boundaries. The utility of the approach is
illustrated using an example from glacier ice petrography.
