Restoration of PSD from chord length distribution data using the method of projections onto convex sets

The interpretation of chord length distributions (CLDs) is essential in many fields and has been discussed by various authors. Here, the technique of the Focused Beam Reflectance Measurement (FBRM) is considered as on-line and in-situ measurement device of the CLD of particle dispersions and emulsions. Though useful in general, this measurement cannot be converted directly into a particle size distribution (PSD), unless the physics of the measurement method is described and accounted for. In this work we present a new tool to carry out such a conversion once the particle shape is known a priori and can be fixed, which is based on a two step procedure: (1) the computation of a matrix that converts the PSD of a population of particles with given shape into the corresponding CLD using a 3-dimensional geometric model; (2) the calculation of the PSD from the resulting linear matrix equation for the measured CLD. Here, the method of Projections onto Convex Sets (POCS) is applied to solve the PSD restoration problem, which is a mathematically ill-posed inverse problem. We study the effect of particle shape and matrix dimension on the ill-posed character of the inverse problem. A detailed error analysis of the CLD allows for a predictive description of a posteriori constraints in the POCS framework. We discuss the application of this method to the characterization of simulated test cases and experimentally obtained data.