Rotation Averaging
Tóm tắt
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.
Tài liệu tham khảo
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).
Afsari, B. (2011). Riemannian \(L^p\) center of mass: Existence, uniqueness, and convexity. Proceedings of the American Mathematical Society, 139(2), 655–673.
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.
Altmann, S. L. (1986). Rotations, quaternions, and double groups. New York: Oxford Science Publications/The Clarendon Press Oxford University Press.
Asgharbeygi, N., & Maleki, A. (2008). Geodesic k-means clustering. In 19th international conference on pattern recognition, ICPR 2008 (pp. 1–4), December 2008.
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.
Beltrami, E. (1868). Teoria fondamentale degli spazii di curvatura costante. Annali di Matematica pura ed Applicata, II (2nd series) (pp. 232–255).
Buchholz, S., & Sommer, G. (2005). On averaging in Clifford groups. Computer Algebra and Geometric Algebra with Applications (pp. 229–238). Berlin: Springer.
Cartan, É. (1951). Leçons sur la géométrie des espaces de Riemann (2nd ed.). Paris: Gauthier-Villars.
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.
Corcuera, J. M., & Kendall, W. S. (1999). Riemannian barycentres and geodesic convexity. Mathematical Proceedings of the Cambridge Philosophical Society, 127, 253–269.
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 .
Daniilidis, K. (1998). Hand-eye calibration using dual quaternions. International Journal of Robotics Research, 18, 286–298.
Devarajan, D., & Radke, R. J. (2007). Calibrating distributed camera networks using belief propagation. EURASIP Journal on Advances in Signal Processing, 1, 2007.
Eckhardt, U. (1980). Weber’s problem and Weiszfeld’s algorithm in general spaces. Mathematical Programming, 18(1), 186–196.
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.
Esquivel, S., Woelk, F., & Koch, R. (2007). Calibration of a multi-camera rig from non-overlapping views. In In DAGM07 (pp. 82–91).
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.
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.
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.
Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society, B, 53(2), 285– 339.
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.
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.
Govindu, V. M. (2006). Robustness in motion averaging. In Proceedings of Asian conference on computer vision (pp. 457–466).
Gramkow, C. (2001). On averaging rotations. International Journal of Computer Vision, 42(1–2), 7–16.
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.
Hartley, R., Aftab, K., & Trumpf, J. (2011). Rotation averaging using the Weiszfeld algorithm. In Proceedings of IEEE conference on computer vision and pattern recognition.
Hartley, R., & Kahl, F. (2009). Global optimization through rotation space search. International Journal of Computer Vision, 82(1), 64–79.
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.
Hartley, R., & Trumpf, J. (2012). Characterization of weakly convex sets in projective space. Technical report, Australian National University.
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).
Hartley, R., & Zisserman, A. (2004). Multiple view geometry in computer vision (2nd ed.). Cambridge: Cambridge University Press.
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.
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.
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.
Hüper, K. (2002). A calculus approach to matrix eigenvalue algorithms. Habilitationsschrift, Universität Würzburg, Germany, July.
Kahl, F. (2005). Multiple view geometry and the \({L}_\infty \)-norm. In Proceedings of international conference on computer vision (pp. 1002–1009).
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.
Kanatani, K. (1990). Group-theoretical methods in image understanding. Berlin: Springer.
Karcher, H. (1977). Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30(5), 509–541.
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.
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.
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.
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.
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.
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.
Le, H. (2001). Locating Fréchet means with application to shape spaces. Advances in Applied Probability, 33, 324–338.
Le, H. (2004). Estimation of Riemannian barycentres. LMS Journal of Computation and Mathematics, 7, 193–200.
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.
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.
Li, Y. (1998). A Newton acceleration of the Weiszfeld algorithm for minimizing the sum of euclidean distances. Computational Optimization and Applications, 10, 219–242.
Lu, F., & Milios, E. (1997). Globally consistent range scan alignment for environment mapping. Autonomous Robots, 4(4), 333–349.
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.
Markley, F., Cheng, Y., Crassidis, J., & Oshman, Y. (2007). Averaging quaternions. Journal of Guidance, Control, and Dynamics, 30(4), 1193–1197.
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.
Massey, W. (1977). Algebraic topology: An introduction. Berlin: Springer.
Moakher, M. (2002). Means and averaging in the group of rotations. SIAM Journal on Matrix Analysis and Applications, 24(1), 1–16.
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.
Morawiec, A. (1998). A note on mean orientation. Journal of Applied Crystallography, 31(5), 818–819.
Morawiec, A. (2004). Orientations and rotations: Computations in crystallographic textures. Berlin: Springer.
Myers, S. (1945). Arcs and geodesics in metric spaces. Transactions of the American Mathematical Society, 57(2), 217–227.
Nocedal, J., & Wright, S. (1999). Numerical optimization. Berlin: Springer.
Ostresh, L. (1978). Convergence of a class of iterative methods for solving weber location problem. Operations Research, 26, 597–609.
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.
Pennec, X. (1998). Computing the mean of geometric features: Application to the mean rotation. Technical Report INRIA RR-3371, INRIA.
Pless, R. (2003). Using many cameras as one. In Proceedings of IEEE conference on computer vision and pattern recognition.
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.
Rinner, B., & Wolf, W. (2008). A bright future for distributed smart cameras. Processings of the IEEE, 96(10), 1562–1564.
Rockafellar, R. (1970). Convex analysis. Princeton, NJ: Princeton University Press.
Rodrigues, R., Barreto, J., & Nunes, U. (2010). Camera pose estimation using images of planar mirror reflections. Computer Vision—ECCV, 2010, 382–395.
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.
Sarlette, A., & Sepulchre, R. (2009). Consensus optimization on manifolds. SIAM Journal on Control and Optimization, 48(1), 56–76.
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.
Steiner, J. (1826). Einige Gesetze über die Theilung der Ebene und des Raumes. Journal für Die Reine Und Angewandte Mathematik, 1, 349–364.
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.
Sturm, P., & Bonfort, T. (2006). How to compute the pose of an object without a direct view? Computer Vision—ACCV, 2006, 21–31.
Subbarao, R., & Meer, P. (2009). Nonlinear mean shift over Riemannian manifolds. International Journal of Computer Vision, 84(1), 1–20.
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.
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.
Weber, A. (1909). Über den Standort der Industrien. Teil 1, Reine Theorie des Standorts. Tübingen: J.C.B. Mohr.
Weiszfeld, E. (1937). Sur le point pour lequel la somme des distances de n points donnes est minimum. Tohoku Mathematical Journal, 43, 355–386.
Wu, F., Wang, Z., & Hu, Z. (2009). Cayley transformation and numerical stability of calibration equation. International Journal of Computer Vision, 82(2), 156–184.
Yang, L. (2010). Riemannian median and its estimation. LMS Journal of Computation and Mathematics, 13, 461–479.
Zhang, H. (1998). Hand/eye calibration for electronic assembly robots. IEEE Transactions on Robotics and Automation, 14(4), 612–616.