A robust inversion method according to a new notion of regularization for poisson data with an application to nanoparticle volume determination

In this paper we present an efficient method for the reconstruction of the volume distribution of diluted polydisperse noninteracting nanoparticles with identical shapes from small angle X-ray scattering measurements. The described method solves a maximum likelihood problem with a positivity constraint on the solution by means of an expectation maximization iterative scheme coupled with a robust stopping criterion. We prove that this is a regularization method according to an innovative notion of regularization specifically defined for inverse problems with Poisson data. Such a regularization, together with an upper bound to the largest retrievable particle size given by the Shannon theorem, results in high fidelity quantitative reconstructions of particle volume distributions, making the method particularly effective in real applications. We test the performance of the method on synthetic data in the case of uni-and bi-modal particle volume distributions. Moreover, we show the reliability of the method on real data provided by a Xenocs device prototype.