Hypersphere Fitting From Noisy Data Using an EM Algorithm

Abstract

This letter studies a new expectation maximization (EM) algorithm to solve the problem of circle, sphere and more generally hypersphere fitting. This algorithm relies on the introduction of random latent vectors having a priori independent von Mises-Fisher distributions defined on the hypersphere. This statistical model leads to a complete data likelihood whose expected value, conditioned on the observed data, has a Von Mises-Fisher distribution. As a result, the inference problem can be solved with a simple EM algorithm. The performance of the resulting hypersphere fitting algorithm is evaluated for circle and sphere fitting.

Deconvolution of spherical data corrupted with unknown noise

“…In [10], the author studies the estimation of the exploration density for noisy circular data on the unit circle (known radius) and with known noise distribution. Comparison of (11) with Theorem 1 in [10] shows that our estimator is rate minimax adaptive to unknown radius, unknown noise distribution and unknown regularity over classes W β (L) for the signal, with a constant deteriorated by a factor at most 2 2β . Comparing (12) with Theorem 2 in [10] shows that a loss in the upper bound for the rate of convergence of the maximum risk of our estimator in case of unknown radius and unknown noise distribution on classes A γ (L) for the signal.…”

“…The statistical estimation of the center and of the radius of the sphere is of interest in various applications such as object tracking, robotics, pattern recognition, see for instance [5], [6], [13], among others, see also [3] and references therein. Several methods have been proposed based on least squares, maximum likelihood, see [11] for a recent likelihood based algorithm, most of them modeling the noise distribution with a Gaussian distribution.…”

Three-Dimensional Whole-Organ Characterization of the Regional Alveolar Morphology in Normal Murine Lungs

“…Parameter quantification was carried out using an in-house code written in Matlab (MathWorks, Version R2017a, Natick, MA, United States). The determination of the alveolar diameter was performed using the Sphere-fit method ( Lesouple et al, 2021 ), which fits spheres within a point cloud using a least-squares algorithm. From the spheres obtained, an active contour algorithm was used to determine the surface and volume of the alveolar cavity ( Strzelecki et al, 2013 ; Aganj et al, 2018 ).…”

Robust Hypersphere Fitting from Noisy Data Using an EM Algorithm

“…Fitting a circle, a sphere or more generally a hypersphere to a noisy point cloud is a recurrent problem in many applications including object tracking [1]- [3], robotics [4]- [6] or image processing and pattern recognition [7]- [9]. This problem was recently investigated in [10] by introducing latent variables defined as affine transformations of random vectors distributed according to von Mises-Fisher distributions. The von Mises-Fisher distribution is a probability distribution defined on the hypersphere and parameterized by a mean vector and a concentration.…”