# Rotation Averaging

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## Abstract

This paper is conceived as a tutorial on rotation averaging, summarizing the research that has been carried out in this area; it discusses methods for single-view and multiple-view rotation averaging, as well as providing proofs of convergence and convexity in many cases. However, at the same time it contains many new results, which were developed to fill gaps in knowledge, answering fundamental questions such as radius of convergence of the algorithms, and existence of local minima. These matters, or even proofs of correctness have in many cases not been considered in the Computer Vision literature. We consider three main problems: single rotation averaging, in which a single rotation is computed starting from several measurements; multiple-rotation averaging, in which absolute orientations are computed from several relative orientation measurements; and conjugate rotation averaging, which relates a pair of coordinate frames. This last is related to the hand-eye coordination problem and to multiple-camera calibration.

## Keywords

Geodesic distance Angular distance Chordal distance Quaternion distance \(L_1\) mean \(L_2\) mean conjugate rotation## Notes

### Acknowledgments

This work was partially supported by NICTA, a research laboratory funded by the Australian Government, in part through the Australian Research Council.

## References

- Absil, P.-A., Mahony, R., & Sepulchre, R. (2008).
*Optimization algorithms on matrix manifolds*. Princeton, NJ: Princeton University Press (With a foreword by Paul Van Dooren).Google Scholar - Afsari, B. (2011). Riemannian \(L^p\) center of mass: Existence, uniqueness, and convexity.
*Proceedings of the American Mathematical Society*,*139*(2), 655–673.MathSciNetMATHCrossRefGoogle Scholar - Agrawal, M. (2006). A Lie algebraic approach for consistent pose registration for general euclidean motion. In
*International conference on intelligent robots and systems*(pp. 1891–1897), October 2006.Google Scholar - Altmann, S. L. (1986).
*Rotations, quaternions, and double groups*. New York: Oxford Science Publications/The Clarendon Press Oxford University Press.MATHGoogle Scholar - Asgharbeygi, N., & Maleki, A. (2008). Geodesic k-means clustering. In
*19th international conference on pattern recognition, ICPR 2008*(pp. 1–4), December 2008.Google Scholar - Baker, P., Fermüller, C., Aloimonos, Y., & Pless, R. (2001). A spherical eye from multiple cameras (makes better models of the world). In
*Proceedings of IEEE conference on computer vision and pattern recognition*(Vol. 1, p. 576). Los Alamitos, CA: IEEE Computer Society.Google Scholar - Beltrami, E. (1868).
*Teoria fondamentale degli spazii di curvatura costante*. Annali di Matematica pura ed Applicata, II (2nd series) (pp. 232–255).Google Scholar - Buchholz, S., & Sommer, G. (2005). On averaging in Clifford groups.
*Computer Algebra and Geometric Algebra with Applications*(pp. 229–238). Berlin: Springer.Google Scholar - Cartan, É. (1951).
*Leçons sur la géométrie des espaces de Riemann*(2nd ed.). Paris: Gauthier-Villars.MATHGoogle Scholar - Clipp, B., Kim, J.-H., Frahm, J.-M., Pollefeys, M., & Hartley, R. (2008). Robust 6DOF motion estimation for non-overlapping multi-camera systems. In
*Workshop on applications of computer vision, WACV08*(pp. 1–8), January 2008.Google Scholar - Corcuera, J. M., & Kendall, W. S. (1999). Riemannian barycentres and geodesic convexity.
*Mathematical Proceedings of the Cambridge Philosophical Society*,*127*, 253–269.MathSciNetMATHCrossRefGoogle Scholar - Dai, Y., Trumpf, J., Li, H., Barnes, N., & Hartley, R. (2009).
*Rotation averaging with application to camera-rig calibration*. In*Proceedings of Asian conference on computer vision*, Xian .Google Scholar - Daniilidis, K. (1998). Hand-eye calibration using dual quaternions.
*International Journal of Robotics Research*,*18*, 286–298.Google Scholar - Devarajan, D., & Radke, R. J. (2007). Calibrating distributed camera networks using belief propagation.
*EURASIP Journal on Advances in Signal Processing*,*1*, 2007.Google Scholar - Eckhardt, U. (1980). Weber’s problem and Weiszfeld’s algorithm in general spaces.
*Mathematical Programming*,*18*(1), 186–196.MathSciNetMATHCrossRefGoogle Scholar - Edelman, A., Arias, T. A., & Smith, S. T. (1998). The geometry of algorithms with orthogonality constraints.
*SIAM Journal on Matrix Analysis and Applications*,*20*(2), 303–353.MathSciNetMATHCrossRefGoogle Scholar - Esquivel, S., Woelk, F., & Koch, R. (2007). Calibration of a multi-camera rig from non-overlapping views. In
*In DAGM07*(pp. 82–91).Google Scholar - Fiori, S., & Tanaka, T. (2008). An averaging method for a committee of special-orthogonal-group machines. In
*IEEE international symposium on circuits and systems, ISCAS 2008*(pp. 2170–2173), May 2008.Google Scholar - Fletcher, P., Lu, C., & Joshi, S. (2003). Statistics of shape via principal geodesic analysis on lie groups. In
*Proceedings of IEEE conference on computer vision and, pattern recognition*(Vol. 1, pp. I-95–I-101), June 2003.Google Scholar - Fletcher, P. T., Venkatasubramanian, S., & Joshi, S. (2009). The geometric median on Riemannian manifolds with applications to robust atlas estimation.
*Neuroimage*,*45*(1 Suppl), 143–152.CrossRefGoogle Scholar - Goodall, C. (1991). Procrustes methods in the statistical analysis of shape.
*Journal of the Royal Statistical Society, B*,*53*(2), 285– 339.MathSciNetMATHGoogle Scholar - Govindu, V. M. (2001). Combining two-view constraints for motion estimation. In
*Proceedings of IEEE conference on computer vision and pattern recognition*(Vol. 2, pp. 218–225). IEEE Computer Society: Los Alamitos, CA.Google Scholar - Govindu, V. M. (2004). Lie-algebraic averaging for globally consistent motion estimation. In
*Proceedings of IEEE conference on computer vision and pattern recognition*(Vol. 1, pp. 684–691). Los Alamitos, CA: IEEE Computer Society.Google Scholar - Govindu, V. M. (2006). Robustness in motion averaging. In
*Proceedings of Asian conference on computer vision*(pp. 457–466).Google Scholar - Gramkow, C. (2001). On averaging rotations.
*International Journal of Computer Vision*,*42*(1–2), 7–16.MathSciNetMATHCrossRefGoogle Scholar - Grove, K., Karcher, H., & Ruh, E. A. (1974). Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems.
*Mathematische Annalen*,*211*, 7–21.MathSciNetMATHCrossRefGoogle Scholar - Hartley, R., Aftab, K., & Trumpf, J. (2011). Rotation averaging using the Weiszfeld algorithm. In
*Proceedings of IEEE conference on computer vision and pattern recognition*.Google Scholar - Hartley, R., & Kahl, F. (2009). Global optimization through rotation space search.
*International Journal of Computer Vision*,*82*(1), 64–79.CrossRefGoogle Scholar - Hartley, R., & Schaffalitzky, F. (2004). \({L}_\infty \) minimization in geometric reconstruction problems. In
*Proceedings of IEEE conference on computer vision and pattern recognition*(pp. I-504–I-509), Washington DC, June 2004.Google Scholar - Hartley, R., & Trumpf, J. (2012). Characterization of weakly convex sets in projective space. Technical report, Australian National University.Google Scholar
- Hartley, R., Trumpf, J., & Dai, Y. (2010). Rotation averaging and weak convexity. In
*Proceedings of the 19th international symposium on mathematical theory of networks and systems (MTNS)*(pp. 2435–2442).Google Scholar - Hartley, R., & Zisserman, A. (2004).
*Multiple view geometry in computer vision*(2nd ed.). Cambridge: Cambridge University Press.MATHCrossRefGoogle Scholar - Horn, B. K. P., Hilden, H., & Negahdaripour, S. (1988). Closed-form solution of absolute orientation using orthonormal matrices.
*Journal of the Optical Society of America*,*5*(7), 1127–1135.MathSciNetCrossRefGoogle Scholar - Humbert, M., Gey, N., Muller, J., & Esling, C. (1996). Determination of a mean orientation from a cloud of orientations. Application to electron back-scattering pattern measurements.
*Journal of Applied Crystallography*,*29*(6), 662–666.CrossRefGoogle Scholar - Humbert, M., Gey, N., Muller, J., & Esling, C. (1998). Response to Morawiec’s (1998) comment on Determination of a mean orientation from a cloud of orientations. Application to electron back-scattering pattern measurements.
*Journal of Applied Crystallography*,*31*(3), 485.CrossRefGoogle Scholar - Hüper, K. (2002).
*A calculus approach to matrix eigenvalue algorithms*. Habilitationsschrift, Universität Würzburg, Germany, July.Google Scholar - Kahl, F. (2005). Multiple view geometry and the \({L}_\infty \)-norm. In
*Proceedings of international conference on computer vision*(pp. 1002–1009).Google Scholar - Kahl, F., & Hartley, R. (2008). Multiple view geometry under the \(L_\infty \)-norm.
*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*30*(9), 1603–1617.CrossRefGoogle Scholar - Kanatani, K. (1990).
*Group-theoretical methods in image understanding*. Berlin: Springer.MATHCrossRefGoogle Scholar - Karcher, H. (1977). Riemannian center of mass and mollifier smoothing.
*Communications on Pure and Applied Mathematics*,*30*(5), 509–541.MathSciNetMATHCrossRefGoogle Scholar - Kaucic, R., Hartley, R., & Dano, N. (2001). Plane-based projective reconstruction. In
*Proceedings of 8th international conference on computer vision*(pp. I-420–I-427), Vancouver, Canada.Google Scholar - Kim, J.-H., Hartley, R., Frahm, J.-M., & Pollefeys, M. (2007). Visual odometry for non-overlapping views using second-order cone programming. In
*Proceedings of Asian conference on computer vision*(Vol. 2, pp. 353–362), November 2007.Google Scholar - Kim, J.-H., Li, H., & Hartley, R. (2008). Motion estimation for multi-camera systems using global optimization. In
*Proceedings of IEEE conference on computer Vision and pattern recognition*.Google Scholar - Kim, J.-H., Li, H., & Hartley, R. (2010). Motion estimation for non-overlapping multi-camera rigs: Linear algebraic and \(L_\infty \) geometric solutions.
*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*32*(6), 1044–1059.CrossRefGoogle Scholar - Krakowski, K., Hüper, K., & Manton, J. (2007). On the computation of the Karcher mean on spheres and special orthogonal groups. In
*RoboMat, workshop on robotics and mathematics*. Portugal: Coimbra.Google Scholar - Kumar, R., Ilie, A., Frahm, J.-M., & Pollefeys, M. (June 2008). Simple calibration of non-overlapping cameras with a mirror. In
*Proceedings of IEEE conference on computer vision and pattern recognition*.Google Scholar - Le, H. (2001). Locating Fréchet means with application to shape spaces.
*Advances in Applied Probability*,*33*, 324–338.MathSciNetMATHCrossRefGoogle Scholar - Le, H. (2004). Estimation of Riemannian barycentres.
*LMS Journal of Computation and Mathematics*,*7*, 193–200.MathSciNetMATHGoogle Scholar - Lébraly, P., Deymier, C., Ait-Aider, O., Royer, E., & Dhome M. (2010). Flexible extrinsic calibration of non-overlapping cameras using a planar mirror: Application to vision-based robotics. In
*2010 IEEE/RSJ International Conference on Intelligent robots and systems (IROS)*(pp. 5640–5647). Taipei: IEEE.Google Scholar - Li, H., Hartley, R., & Kim, J.-H. (2008). Linear approach to motion estimation using generalized camera models. In
*Proceeding of IEEE conference on computer vision and pattern recognition*.Google Scholar - Li, Y. (1998). A Newton acceleration of the Weiszfeld algorithm for minimizing the sum of euclidean distances.
*Computational Optimization and Applications*,*10*, 219–242.MathSciNetMATHCrossRefGoogle Scholar - Lu, F., & Milios, E. (1997). Globally consistent range scan alignment for environment mapping.
*Autonomous Robots*,*4*(4), 333–349.CrossRefGoogle Scholar - Manton, J. H. (2004). A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups. In
*Proceedings of the eighth international conference on control, automation, robotics and vision*(pp. 2211–2216), Kunming, China, December 2004.Google Scholar - Markley, F., Cheng, Y., Crassidis, J., & Oshman, Y. (2007). Averaging quaternions.
*Journal of Guidance, Control, and Dynamics*,*30*(4), 1193–1197.CrossRefGoogle Scholar - Martinec, D., & Pajdla, T. (June 2007). Robust rotation and translation estimation in multiview reconstruction. In
*Proceedings of IEEE conference on computer vision and pattern recognition*.Google Scholar - Massey, W. (1977).
*Algebraic topology: An introduction*. Berlin: Springer.Google Scholar - Moakher, M. (2002). Means and averaging in the group of rotations.
*SIAM Journal on Matrix Analysis and Applications*,*24*(1), 1–16.MathSciNetMATHCrossRefGoogle Scholar - Morawiec, A. (1998). Comment on Determination of a mean orientation from a cloud of orientations. Application to electron back-scattering pattern measurements by Humbert et al. (1996).
*Journal of Applied Crystallography*,*31*(3), 484.CrossRefGoogle Scholar - Morawiec, A. (1998). A note on mean orientation.
*Journal of Applied Crystallography*,*31*(5), 818–819.CrossRefGoogle Scholar - Morawiec, A. (2004).
*Orientations and rotations: Computations in crystallographic textures*. Berlin: Springer.CrossRefGoogle Scholar - Myers, S. (1945). Arcs and geodesics in metric spaces.
*Transactions of the American Mathematical Society*,*57*(2), 217–227.MathSciNetMATHCrossRefGoogle Scholar - Nocedal, J., & Wright, S. (1999).
*Numerical optimization*. Berlin: Springer.MATHCrossRefGoogle Scholar - Ostresh, L. (1978). Convergence of a class of iterative methods for solving weber location problem.
*Operations Research*,*26*, 597–609.MathSciNetMATHCrossRefGoogle Scholar - Park, F., & Martin, B. (1994). Robot sensor calibration: solving AX=XB on the euclidean group.
*IEEE Transactions on Robotics and Automation*,*10*(5), 717–721.CrossRefGoogle Scholar - Pennec, X. (1998). Computing the mean of geometric features: Application to the mean rotation. Technical Report INRIA RR-3371, INRIA.Google Scholar
- Pless, R. (2003). Using many cameras as one. In
*Proceedings of IEEE conference on computer vision and pattern recognition*.Google Scholar - Qi, C., Gallivan, K. A., & Absil, P.-A. (2010). Riemannian BFGS algorithm with applications. In M. Diehl, F. Glineur, E. Jarlebring, & W. Michiels (Eds.),
*Recent advances in optimization and its applications in engineering*(pp. 183–192). Berlin: Springer.CrossRefGoogle Scholar - Rinner, B., & Wolf, W. (2008). A bright future for distributed smart cameras.
*Processings of the IEEE*,*96*(10), 1562–1564.CrossRefGoogle Scholar - Rockafellar, R. (1970).
*Convex analysis*. Princeton, NJ: Princeton University Press.MATHGoogle Scholar - Rodrigues, R., Barreto, J., & Nunes, U. (2010). Camera pose estimation using images of planar mirror reflections.
*Computer Vision—ECCV*,*2010*, 382–395.Google Scholar - Rother, C., & Carlsson, S. (2001). Linear multi view reconstruction and camera recovery. In
*Proceedings of 8th international conference on computer vision*(pp. I-42–I-49), Vancouver, Canada.Google Scholar - Sarlette, A., & Sepulchre, R. (2009). Consensus optimization on manifolds.
*SIAM Journal on Control and Optimization*,*48*(1), 56–76.Google Scholar - Sim, K., & Hartley, R. (2006). Recovering camera motion using \({L}_{\infty }\) minimization. In
*Proceedings of IEEE conference on computer vision and pattern recognition*, New York City.Google Scholar - Steiner, J. (1826). Einige Gesetze über die Theilung der Ebene und des Raumes.
*Journal für Die Reine Und Angewandte Mathematik*,*1*, 349–364.MATHCrossRefGoogle Scholar - Strobl, K., & Hirzinger, G. (2006) . Optimal hand-eye calibration. In
*2006 IEEE/RSJ international conference on intelligent robots and systems*(pp. 4647–4653), October 2006.Google Scholar - Sturm, P., & Bonfort, T. (2006). How to compute the pose of an object without a direct view?
*Computer Vision—ACCV*,*2006*, 21–31.Google Scholar - Subbarao, R., & Meer, P. (2009). Nonlinear mean shift over Riemannian manifolds.
*International Journal of Computer Vision*,*84*(1), 1–20. Google Scholar - Teller, S., Antone, M., Bodnar, Z., Bosse, M., Coorg, S., Jethwa, M., et al. (2003). Calibrated, registered images of an extended urban area.
*International Journal of Computer Vision*,*53*(1), 93–107.Google Scholar - Tron, R., Vidal, R., & Terzis, A. (2008). Distributed pose averaging in camera networks via consensus on SE(3). In
*Second ACM/IEEE international conference on distributed smart cameras*, September 2008.Google Scholar - Weber, A. (1909).
*Über den Standort der Industrien. Teil 1, Reine Theorie des Standorts*. Tübingen: J.C.B. Mohr.Google Scholar - Weiszfeld, E. (1937). Sur le point pour lequel la somme des distances de n points donnes est minimum.
*Tohoku Mathematical Journal*,*43*, 355–386.Google Scholar - Wu, F., Wang, Z., & Hu, Z. (2009). Cayley transformation and numerical stability of calibration equation.
*International Journal of Computer Vision*,*82*(2), 156–184.Google Scholar - Yang, L. (2010). Riemannian median and its estimation.
*LMS Journal of Computation and Mathematics*,*13*, 461–479.Google Scholar - Zhang, H. (1998). Hand/eye calibration for electronic assembly robots.
*IEEE Transactions on Robotics and Automation*,*14*(4), 612–616.CrossRefGoogle Scholar