L1 rotation averaging using the Weiszfeld algorithm

We consider the problem of rotation averaging under the L1 norm. This problem is related to the classic Fermat-Weber problem for finding the geometric median of a set of points in IRn. We apply the classical Weiszfeld algorithm to this problem, adapting it iteratively in tangent spaces of SO(3) to obtain a provably convergent algorithm for finding the L1 mean. This results in an extremely simple and rapid averaging algorithm, without the need for line search. The choice of L1 mean (also called geometric median) is motivated by its greater robustness compared with rotation averaging under the L2 norm (the usual averaging process). We apply this problem to both single-rotation averaging (under which the algorithm provably finds the global L1 optimum) and multiple rotation averaging (for which no such proof exists). The algorithm is demonstrated to give markedly improved results, compared with L2 averaging. We achieve a median rotation error of 0.82 degrees on the 595 images of the Notre Dame image set.

[1]  Suresh Venkatasubramanian,et al.  The geometric median on Riemannian manifolds with application to robust atlas estimation , 2009, NeuroImage.

[2]  Venu Madhav Govindu.  Lie-algebraic averaging for globally consistent motion estimation , 2004, CVPR 2004.

[3]  K. Hüper,et al.  On the Computation of the Karcher Mean on Spheres and Special Orthogonal Groups , 2007 .

[4]  M. Shirosaki.  Another proof of the defect relation for moving targets , 1991 .

[5]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[6]  Richard Szeliski,et al.  Modeling the World from Internet Photo Collections , 2008, International Journal of Computer Vision.

[7]  H. Karcher,et al.  Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems , 1974 .

[8]  Maher Moakher,et al.  To appear in: SIAM J. MATRIX ANAL. APPL. MEANS AND AVERAGING IN THE GROUP OF ROTATIONS∗ , 2002 .

[9]  Ulrich Eckhardt,et al.  Weber's problem and weiszfeld's algorithm in general spaces , 1980, Math. Program..

[10]  Tomás Pajdla,et al.  Robust Rotation and Translation Estimation in Multiview Reconstruction , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[11]  David Nistér,et al.  An efficient solution to the five-point relative pose problem , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[12]  Richard I. Hartley,et al.  Multiple-View Geometry Under the {$L_\infty$}-Norm , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  R. Hartley,et al.  L/sub /spl infin// minimization in geometric reconstruction problems , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[14]  B. Afsari.  Riemannian Lp center of mass: existence, uniqueness, and convexity , 2011 .

[15]  Manuel Blum,et al.  Time Bounds for Selection , 1973, J. Comput. Syst. Sci..

[16]  Carsten Rother,et al.  Linear Multi View Reconstruction and Camera Recovery , 2001, ICCV.

[17]  Andrew Zisserman,et al.  Multiple View Geometry in Computer Vision (2nd ed) , 2003 .

[18]  Le Yang.  Riemannian Median and Its Estimation , 2009, 0911.3474.

[19]  Robert Kaucic,et al.  Plane-based projective reconstruction , 2001, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001.

[20]  Venu Madhav Govindu,et al.  Combining two-view constraints for motion estimation , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[21]  Jonathan H. Manton,et al.  A globally convergent numerical algorithm for computing the centre of mass on compact Lie groups , 2004, ICARCV 2004 8th Control, Automation, Robotics and Vision Conference, 2004..

[22]  Steven M. Seitz,et al.  Photo tourism: exploring photo collections in 3D , 2006, ACM Trans. Graph..

[23]  HighWire Press.  Philosophical Transactions of the Royal Society of London , 1781, The London Medical Journal.

[24]  Yuying Li,et al.  A Newton Acceleration of the Weiszfeld Algorithm for Minimizing the Sum of Euclidean Distances , 1995, Comput. Optim. Appl..

[25]  Richard I. Hartley,et al.  Recovering Camera Motion Using L\infty Minimization , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[26]  Lawrence M. Ostresh.  On the Convergence of a Class of Iterative Methods for Solving the Weber Location Problem , 1978, Oper. Res..

[27]  H. Opower.  Multiple view geometry in computer vision , 2002 .

[28]  Alain Sarlette,et al.  Consensus Optimization on Manifolds , 2008, SIAM J. Control. Optim..

[29]  Richard Hartley,et al.  Minimizing algebraic error , 1998, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[30]  Nick Barnes,et al.  Rotation Averaging with Application to Camera-Rig Calibration , 2009, ACCV.

[31]  F. Kahl.  Multiple View Geometry and the L-infinity Norm , 2005, ICCV 2005.