Sparse Bayesian learning for the Laplace transform inversion in dynamic light scattering

Abstract

A new method is described using the sparse Bayesian learning (SBL) algorithm of Tipping to obtain an optimal and reliable solution to the Laplace transform inversion in dynamic light scattering (DLS). The linear inverse problem in DLS has numerical solutions that depend on their domains and dimensions. For a given domain and dimension, a sparse solution in an SBL framework is the most-probable solution and can be used for classifying a system of objects by a few relevant values. Recently, we have shown that the SBL algorithm of Tipping is suitable for studying cataract in ocular lenses by describing the opacity of a lens with a few dominant sizes of crystallin proteins in the lens. However, since the sparseness of SBL solutions cannot reflect a true system, we need to develop a method by using the SBL algorithm to give a true description and, at the same time, a useful classification of the opacity of lenses. We generate a set of sparse solutions of different domains but of the same dimension, and then superimpose them to give a general solution with its dimension treated as a regularization parameter. An optimal solution, which provides a reliable description of a particle system, is determined by the L-curve criterion for selecting the suitable value of the regularization parameter. The performance of our method is evaluated by analyzing simulated data generated from unimodal and bimodal distributions. From the reconstructed distributions, we see that our method gives high resolution comparable to the sophisticated Bryan’s maximum-entropy algorithm, which gives better resolution than CONTIN. Our method is then applied to experimental DLS data of the ocular lenses of a fetal calf and a Rhesus monkey to obtain optimal particle size distributions of crystallins and the crystallin aggregates in the lenses. We conclude by discussing possible improvements on our method for analyzing DLS data and for solving any linear inverse problem by an SBL algorithm.