Numerical reconstruction of two-dimensional particle size distributions from laser diffraction data

Abstract

In this paper, we consider the two-dimensional Fredholm integral equation of the first kind. The kernel of this equation models the scattering of a laser beam by spheroids having the same fixed orientation. The unknown function under the integral describes the distribution of spheroids along two semi-axes. The input data are the diffraction pattern corresponding to the scattering of the laser beam by the particles. We show that the Tikhonov regularization method allows one to reconstruct two-dimensional distributions in the case when the diffraction pattern is modulated by white noise with relative amplitude up to 1%. In applications, this means novel possibility of obtaining two-dimensional particle size distributions, rather than one-dimensional ones, as in the classical version of the method. This significantly expands the capabilities of particle sizing via static laser diffraction technique. We show that the solution of the corresponding equation is unique in space and exists when the right-hand-side function is in a set dense in . We provide test experimental results in the framework of laser ektacytometry of red blood cells, which serves as the main application of the proposed approach presently.