When is Rotations Averaging Hard?

Rotations averaging has become a key subproblem in global Structure from Motion methods. Several solvers exist, but they do not have guarantees of correctness. They can produce high-quality results, but also sometimes fail. Our understanding of what makes rotations averaging problems easy or hard is still very limited. To investigate the difficulty of rotations averaging, we perform a local convexity analysis under an \(L_2\) cost function. Although a previous result has shown that in general, this problem is locally convex almost nowhere, we show how this negative conclusion can be reversed by considering the gauge ambiguity. Our theoretical analysis reveals the factors that determine local convexity—noise and graph structure—as well as how they interact, which we describe by a particular Laplacian matrix. Our results are useful for predicting the difficulty of problems, and we demonstrate this on practical datasets. Our work forms the basis of a deeper understanding of the key properties of rotations averaging problems, and we discuss how it can inform the design of future solvers for this important problem.

[1]  René Vidal,et al.  Intrinsic consensus on SO(3) with almost-global convergence , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[2]  Andrea Fusiello,et al.  Hierarchical structure-and-motion recovery from uncalibrated images , 2015, Comput. Vis. Image Underst..

[3]  Amit Singer,et al.  Robust estimation of rotations from relative measurements by maximum likelihood , 2013, 52nd IEEE Conference on Decision and Control.

[4]  Hongdong Li,et al.  Rotation Averaging , 2013, International Journal of Computer Vision.

[5]  Amit Singer,et al.  Exact and Stable Recovery of Rotations for Robust Synchronization , 2012, ArXiv.

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

[7]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[8]  Onur Özyesil,et al.  Robust camera location estimation by convex programming , 2014, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[9]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[10]  Andrew Owens,et al.  Discrete-continuous optimization for large-scale structure from motion , 2011, CVPR 2011.

[11]  Noah Snavely,et al.  Robust Global Translations with 1DSfM , 2014, ECCV.

[12]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[13]  Ping Tan,et al.  Global Structure-from-Motion by Similarity Averaging , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[14]  Tobias Höllerer,et al.  Optimizing the Viewing Graph for Structure-from-Motion , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[15]  Amit Singer,et al.  A Cheeger Inequality for the Graph Connection Laplacian , 2012, SIAM J. Matrix Anal. Appl..

[16]  Johan Fredriksson,et al.  Simultaneous Multiple Rotation Averaging Using Lagrangian Duality , 2012, ACCV.

[17]  金谷 健一.  Group-theoretical methods in image understanding , 1990 .

[18]  Andreas Terzis,et al.  Distributed pose averaging in camera networks via consensus on SE(3) , 2008, 2008 Second ACM/IEEE International Conference on Distributed Smart Cameras.

[19]  Jochen Trumpf,et al.  L1 rotation averaging using the Weiszfeld algorithm , 2011, CVPR 2011.

[20]  B. Rossi,et al.  Robust Absolute Rotation Estimation via Low-Rank and Sparse Matrix Decomposition , 2014, 2014 2nd International Conference on 3D Vision.

[21]  Vincent D. Blondel,et al.  Cramér-Rao bounds for synchronization of rotations , 2012, ArXiv.

[22]  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.

[23]  Frank Dellaert,et al.  Spectral partitioning for structure from motion , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[24]  Venu Madhav Govindu,et al.  Efficient and Robust Large-Scale Rotation Averaging , 2013, 2013 IEEE International Conference on Computer Vision.

[25]  Venu Madhav Govindu,et al.  Lie-algebraic averaging for globally consistent motion estimation , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..