Determination of Particle-Size Distributions from Light-Scattering Measurement Using Constrained Gaussian Process Regression

TL;DR

This paper introduces a robust, constrained Gaussian process regression method for accurately estimating particle size distributions from optical scattering data, effectively addressing the ill-posed inverse problem with improved stability and physical interpretability.

Contribution

The work presents a novel constrained Gaussian process framework with spectral kernel expansion for stable, efficient particle size distribution estimation from scattering measurements, incorporating physical constraints.

Findings

Accurately reconstructs particle size distributions from noisy data.

Produces stable, smooth, and physically interpretable results.

Enhances computational efficiency with spectral kernel expansion.

Abstract

In this work, we propose a novel methodology for robustly estimating particle size distributions from optical scattering measurements using constrained Gaussian process regression. The estimation of particle size distributions is commonly formulated as a Fredholm integral equation of the first kind, an ill-posed inverse problem characterized by instability due to measurement noise and limited data. To address this, we use a Gaussian process prior to regularize the solution and integrate a normalization constraint into the Gaussian process via two approaches: by constraining the Gaussian process using a pseudo-measurement and by using Lagrange multipliers in the equivalent optimization problem. To improve computational efficiency, we employ a spectral expansion of the covariance kernel using eigenfunctions of the Laplace operator, resulting in a computationally tractable low-rank representation without sacrificing accuracy. Additionally, we investigate two complementary strategies for hyperparameter estimation: a data-driven approach based on maximizing the unconstrained log marginal likelihood, and an alternative approach where the physical constraints are taken into account. Numerical experiments demonstrate that the proposed constrained Gaussian process regression framework accurately reconstructs particle size distributions, producing numerically stable, smooth, and physically interpretable results. This methodology provides a principled and efficient solution for addressing inverse scattering problems and related ill-posed integral equations.