Bayesian Inference of a Parametric Random Spheroid from its Orthogonal Projections

Abstract

The paper focuses on a new method for the inference of a parametric random spheroid from the observations of its 2D orthogonal projections. Such a stereological problem is well-known from the literature when the projections come from only one deterministic spheroid. Nevertheless, when the spheroid is random itself, the estimation of its distribution is not straightforward. From a theoretical viewpoint, it is shown that the semi-axes of the spheroid and the ones of the projected ellipses are linked through a random polynomial of degree two which admits two real random positive roots. The likelihood can be formulated in terms of the coefficients of the random polynomial, but is not analytically tractable. Assuming that the random spheroid is parameterized by a set of parameters θ real , an approximation of the maximum a posteriori is used to estimate θ real. The estimator is based on the so-called approximate Bayesian computation method and a kernel density technique. As an illustration, the case of a spheroids population, whose semi-major axis follows a gamma distribution and the flattening coefficient a truncated normal distribution, is studied. The numerical results demonstrate that the bias of the estimator is very low, with a reasonable variance, both for the first and the second order moments of the semi-axes. The proposed method enables to recover some 3D morphological characteristics of a population of independent and

Estimation of 3D geometrical properties of spheroid-like particle systems using projection images

“…This equivalence is due to (A1) and to the i.i.d hypothesis for the particles. In [16], we proved that the likelihood function L(. ; y) is not analytically tractable, hence, the posterior density cannot be easily maximized.…”

“…. , θ n ) from f (•|ρ(S(y ), S(y)) ≤ δ) (see [16] for practical consideration of the sampling algorithm), a kernel density estimator of f (•|ρ(S(y ), S(y)) ≤ δ) is obtained with…”

“…Finally, the main steps of the algorithm is summarized in Algorithm 1. For the computation of the bandwidth matrix H, see [16].…”

“…Several segmentation algorithms have been proposed in the literature for overlapping ellipses in 2D images, see for example [8], [10]- [13]. However, very few stereological methods exist to infer the geometrical properties of the system from the projected ellipses [16].…”

Estimating the Parameters of a Stochastic Geometrical Model for Multiphase Flow Images Using Local Measures

“…A cost function is then used to compare these characteristics with the values observed in the real images, and the best set of parameter values for the stochastic model is estimated using an optimization process. The main problem with this approach is that although these models are usually able to capture some of the complexity of real flows (De Langlard et al (2021); Kracht et al (2013)), they can be difficult to optimize, especially for more complex flows requiring a greater number of parameters.…”