Statistical methods for random rotations

Abstract

Data as three-dimensional rotations have application in computer science, kinematics and materials sciences, among other areas. Estimating the central orientation from a sample of such data is an important problem, which is complicated by the fact that several different approaches exist for this, motivated by various geometrical and decision-theoretic considerations. However, little is known about how such estimators compare, especially on common distributions for location models with random rotations. We examine four location estimators, three of which are commonly found in different literatures and the fourth estimator (a projected median) is newly introduced. Our study unifies existing literature and provides a detailed numerical investigation of location estimators for three commonly used rotation distributions in statistics and materials science. While the data-generating model influences the best choice of an estimator, the proposed projected median emerges as an overall good performer, which can be suggested without particular distributional assumptions. We illustrate the estimators and our findings with data from a materials science study by approximating the central orientation of cubic crystals on the micro-surface of a metal. Accompanying supplementary materials are available online.

Approximate Differentiable Rendering with Algebraic Surfaces

“…However, for pose estimation, we exclude its results from the overall average due to symmetry. Our poses are generated with rotations of uniform axis and angle uniformly between -45 and 45 degrees (uniform-axis random spin [63]). The gear model however has 15 teeth and a rotational symmetry of 24 degrees when viewed from one side, as seen in Fig.…”