Abstract
3D rotations arise in many computer vision, computer graphics, and robotics problems and evaluation of the distance between two 3D rotations is often an essential task. This paper presents a detailed analysis of six functions for measuring distance between 3D rotations that have been proposed in the literature. Based on the well-developed theory behind 3D rotations, we demonstrate that five of them are bi-invariant metrics on SO(3) but that only four of them are boundedly equivalent to each other. We conclude that it is both spatially and computationally more efficient to use quaternions for 3D rotations. Lastly, by treating the two rotations as a true and an estimated rotation matrix, we illustrate the geometry associated with iso-error measures.
Similar content being viewed by others
References
Craig, J.J.: Introduction to Robotics: Mechanics and Control. Addison Wesley, Reading (1986)
Deutscher, J., Davison, A.J., Reid, I.: Automatic partitioning of high dimensional search spaces associated with articulated body motion capture. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition, vol. 2, pp. 669–676 (2001)
Faugeras, O.D., Toscani, G.: The calibration problem for stereo. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 15–20. Miami Beach, Florida, USA (1986)
Heeger, D.J., Jepson, A.: Simple method for computing 3D motion and depth. In: Proc. International Conference on Computer Vision, pp. 96–100 (1990)
Horn, B.K.P.: Closed form solution of absolute orientation using unit quaternions. J. Opt. Soc. Am. 4(4), 629–642 (1987)
Huynh, D.Q., Heyden, A.: Scene point constraints in camera auto-calibration: an implementational perspective. Image Vis. Comput. 23(8), 747–760 (2005)
Kelly, J.L.: General Topology. Van Nostrand, New York (1955)
Knight, J., Reid, I.: Automated alignment of robotic pan-tilt camera units using vision. Int. J. Comput. Vis. 68(3), 219–237 (2006)
Kuffner, J.J.: Effective sampling and distance metrics for 3D rigid body path planning. In: Proc. International Conference on Robotics and Automation (2004)
Larochelle, P.M., Murray, A.P., Angeles, J., A distance metric for finite sets of rigid-body displacement in the polar decomposition. ASME J. Mech. Des. 129, 883–886 (2007)
McCarthy, J.M.: An Introduction to Theoretical Kinematics. MIT Press, Cambridge (1990)
Munkres, J.R.: Topology: A First Course. Prentice-Hall, New York (1975)
Park, F.C.: Distance metrics on the rigid-body motions with applications to mechanism design. ASME J. Mech. Des. 117, 48–54 (1995)
Park, F.C., Ravani, B.: Smooth invariant interpolation of rotations. ACM Trans. Graph. 16(3), 277–295 (1997)
Pervin, E., Webb, J.: Quaternions in computer vision and robotics. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 382–383. Los Alamitos, CA (1983)
Ravani, B., Roth, B.: Motion synthesis using kinematic mappings. ASME J. Mech. Transm. Autom. Des. 105, 460–467 (1983)
Shoemake, K.: Animating rotation with quaternion curves. SIGGRAPH 19(3), 245–254 (1985)
Watt, A., Watt, M.: Advanced Animation and Rendering Techniques. Addison Wesley, Reading (1992)
Wolfram MathWorld. http://mathworld.wolfram.com/EulerAngles.html
Wunsch, P., Winkler, S., Hirzinger, G.: Real-time pose estimation of 3D objects from camera images using neural networks. In: Proc. International Conference on Robotics and Automation, vol. 4, pp. 3232–3237 (1997)
Zisserman, A., Beardsley, P.A., Reid, I.: Metric calibration of a stereo rig. In: Proc. IEEE Workshop on Representations of Visual Scenes, pp. 93–100. Boston, USA (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was in part supported by a UWA study leave grant.
Rights and permissions
About this article
Cite this article
Huynh, D.Q. Metrics for 3D Rotations: Comparison and Analysis. J Math Imaging Vis 35, 155–164 (2009). https://doi.org/10.1007/s10851-009-0161-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-009-0161-2