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 identically distributed spheroids thanks to the only observations of its projected ellipses.

本文着重从其二维正交投影的观察结果推论参数随机球体的新方法。当投影仅来自一个确定性球体时，这种立体问题在文献中是众所周知的。然而，当球体本身是随机的时，对其分布的估计并不简单。从理论上看，球体的半轴和投影椭圆的半轴通过一个二阶随机多项式链接，该多项式允许两个真实的随机正根。可能性可以根据随机多项式的系数来表述，但在分析上不易处理。假定随机球体由一组参数的参数θ_实中，最大的近似后验被用于估计θ_真实。估计器基于所谓的近似贝叶斯计算方法和核密度技术。作为说明，研究了一个球体总体的情况，该球体的半长轴遵循伽马分布，而展平系数则为截断的正态分布。数值结果表明，对于半轴的第一阶和第二阶矩，估计器的偏差非常低，并且具有合理的方差。所提出的方法能够恢复一群独立且均匀分布的椭球体的某些3D形态特征，这要归功于其投影椭圆的唯一观测结果。