Ceramics-Silikáty 61, (2) 147 - 157 (2017)

A generalized formulation of transformation matrices is given for the reconstruction of sphere diameter distributions from their section circle diameter distributions. This generalized formulation is based on a weight shift parameter that can be adjusted from 0 to 1. It includes the well-known Saltykov and Cruz-Orive transformations as special cases (for parameter values of 0 and 0.5, respectively). The physical meaning of this generalization is explained (showing, among others, that the Woodhead transformation should be bounded by the Saltykov transformation on the one side and by our transformation from the other) and its numerical performance is investigated. In particular, it is shown that our generalized transformation is numerically highly unstable, i.e. introduces numerical artefacts (oscillations or even unphysical negative sphere frequencies) into the reconstruction, and can lead to completely wrong results when a critical value of the parameter (usually in the range 0.7-0.9, depending on the type of distribution) is exceeded. It is shown that this numerical instability is an intrinsic feature of these transformations that depends not only on the weight shift parameter value and is affected both by the type and the position of the distribution. It occurs in a natural way also for the Cruz-Orive and other transformations with finite weight shift parameter values and is not just caused by inadequate input data (e.g. as a consequence of an insufficient number of objects counted), as commonly assumed. Finally it is shown that an even more general class of transformation matrices can be defined that includes, in addition to the aformentioned transformations, also the Wicksell transformation.