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  • Open Access

Determination of particle-size distributions from light-scattering measurement using constrained Gaussian process regression

Fahime Seyedheydari*, Mahdi Nasiri, Marcin Mińkowski, and Simo Särkkä

  • Department of Electrical Engineering and Automation (EEA), Aalto University, Espoo, Finland

  • *Contact author: fahime.seyedheydari@aalto.fi
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Phys. Rev. E 113, 024120 – Published 13 February, 2026

DOI: https://doi.org/10.1103/91rp-syk9

Abstract

In this work, we propose a 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 pseudomeasurement 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.

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Phys. Rev. E 113, 024120– Published 13 February, 2026
  • Received 14 August 2025
  • Revised 6 November 2025
  • Accepted 13 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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