Estimating 3-D rigid body transformations: a comparison of four major algorithms

A common need in machine vision is to compute the 3-D rigid body transformation that aligns two sets of points for which correspondence is known. A comparative analysis is presented here of four popular and efficient algorithms, each of which computes the translational and rotational components of the transform in closed form, as the solution to a least squares formulation of the problem. They differ in terms of the transformation representation used and the mathematical derivation of the solution, using respectively singular value decomposition or eigensystem computation based on the standard \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $[ \vec{R}, \vec{T} ]$\end{document} representation, and the eigensystem analysis of matrices derived from unit and dual quaternion forms of the transform. This comparison presents both qualitative and quantitative results of several experiments designed to determine (1) the accuracy and robustness of each algorithm in the presence of different levels of noise, (2) the stability with respect to degenerate data sets, and (3) relative computation time of each approach under different conditions. The results indicate that under “ideal” data conditions (no noise) certain distinctions in accuracy and stability can be seen. But for “typical, real-world” noise levels, there is no difference in the robustness of the final solutions (contrary to certain previously published results). Efficiency, in terms of execution time, is found to be highly dependent on the computer system setup.