Bayesian inference of a parametric random ellipsoid from its orthogonal projections

Mathieu de Langlard (1) , Fabrice Lamadie (1) , Sophie Charton (1) , Johan Debayle (2, 3)

1 DMRC - Département de recherche sur les procédés pour la mine et le recyclage du combustible
2 LGF-ENSMSE - Laboratoire Georges Friedel
3 PMMG-ENSMSE - Département Procédés de Mise en oeuvre des Milieux Granulaires

This article focuses on a new method for the inference of a parametric random ellipsoid from the observations of its 2D orthogonal projections. Such a stereological problem is well-know from the literature when the random projections comes from only one ellipsoid. Nevertheless, when the ellipsoid is random itself, the estimation of its distribution is not straightforward. From a theoretical viewpoint, we show that the semi-axes of the ellipsoid and the ones of the projected ellipses are linked through a random polynomial of degree two. Assuming that the ellipsoid is actually a spheroid, it is shown that the random polynomial admits two random real positive roots. The likelihood can then be formulated in terms of the coefficients of the random polynomial. As these coefficients are non-linear functions of the semi-axes of the spheroid itself, the likelihood is not tractable in general. Therefore, an approximation of the posterior distribution is here proposed which is based on a Markov chain Monte Carlo version of the Approximate Bayesian Computation (ABC) method. The numerical results highlight that this technique leads to low errors (less than 10%) for the estimation of the first and second order moments of the semi-axes of the spheroid. Moreover, the required number of observations does not need necessarily to be important: around 500 observed projected ellipses turns out to achieve low errors. The proposed method enables to recover some 3D morphological characteristics of a population of independent and identically distributed spheroids thanks to the only observations of its projected ellipses.

Mots clés: Bayesian inference; Random spheroid; Orthogonal projection; Stereology; Morphological characterization; Approximate Bayesian computation; Markov chain Monte Carlo method

Domaines: Traitement des images [eess.IV]; Probabilités [math.PR]; Traitement du signal et de l'image [eess.SP]

Mathieu de Langlard, Fabrice Lamadie, Sophie Charton, Johan Debayle. Bayesian inference of a parametric random ellipsoid from its orthogonal projections. Methodology and Computing in Applied Probability, 2021, 23, pp.549 à 567. ⟨10.1007/s11009-020-09806-w⟩. ⟨hal-02927445⟩