Abstract
We consider the problem of recovering individual motions for a set of cameras when we are given a number of relative motion estimates between camera pairs. Typically, the number of such relative motion pairs exceeds the number of unknown camera motions, resulting in an overdetermined set of relationships that needs to be averaged. This problem occurs in a variety of contexts and in this chapter we consider sensor localization in 3D reconstruction problems where the camera motions belong to specific finite-dimensional Lie groups or motion groups. The resulting problem has a rich geometric structure that leads to efficient and accurate algorithms that are also robust to the presence of outliers. We develop the motion averaging framework and demonstrate its utility in 3D motion estimation using images or depth scans. A number of real-world examples exemplify the motion averaging principle as well as elucidate its advantages over conventional approaches for 3D reconstruction.
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References
Agarwal S, Snavely N, Simon I, Seitz S, Szeliski R (2009) Building rome in a day. In: Proceedings of the international conference on computer vision, pp 72–79
Benjemaa R, Schmitt F (1997) Fast global registration of 3d sampled surfaces using a multi-z-buffer technique. In: 3-D digital imaging and modeling, (3DIM), pp 113–120
Boothby WM (2003) An introduction to differentiable manifolds and riemannian geometry, revised 2nd edn. Academic, New York
Candes E, Tao T (2005) Decoding by linear programming. IEEE Trans Inf Theory 51(12):4203–4215
Chatterjee A, Govindu VM (2013) Efficient and robust large-scale rotation averaging. In:Â IEEE international conference on computer vision (ICCV)
Chatterjee A, Jain S, Govindu VM (2012) A pipeline for building 3d models using depth cameras. In:Â Proceedings of the eighth indian conference on computer vision, graphics and image processing
Crandall DJ, Owens A, Snavely N, Huttenlocher D (2011) Discrete-continuous optimization for large-scale structure from motion. In:Â CVPR
Fischler MA, Bolles RC (1981) Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM 24(6):381–395
Govindu VM (2001) Combining two-view constraints for motion estimation. In:Â CVPR
Govindu VM (2004) Lie-algebraic averaging for globally consistent motion estimation. In:Â CVPR
Govindu VM (2006) Robustness in motion averaging. In:Â Asian conference on computer vision (ACCV)
Govindu VM, Pooja A (2014) On averaging multiview relations for 3d scan registration. IEEE Trans Image Process 23(3):1289–1302
Hartley RI, Zisserman A (2004) Multiple view geometry in computer vision, 2nd edn. Cambridge University Press, Cambridge
Hartley RI, Aftab K, Trumpf J (2011) L1 rotation averaging using the weiszfeld algorithm. In:Â CVPR
Hartley RI, Trumpf J, Dai Y, Li H (2013) Rotation averaging. Int J Comput Vis 103(3):267–305
Kanatani K (1990) Group theoretic methods in image understanding. Springer, Heidelberg
Kanatani K (1993) Geometric computation for machine vision. Oxford University Press, Oxford
Karcher H (1997) Riemannian center of mass and mollifier smoothing. Commun Pure Appl Math 30(5):509–541
Krakowski K, Huper K, Manton J (2007) On the computation of the karcher mean on spheres and special orthogonal groups. In: Workshop on robotics and mathematics, RoboMat ’07
Manton JH (2004) A globally convergent numerical algorithm for computing the centre of mass on compact lie groups. In: International conference on control, automation, robotics and vision, ICARCV 2004, pp 2211–2216
Moulon P, Monasse P, Marlet R (2013) Global fusion of relative motions for robust, accurate and scalable structure from motion. In: IEEE international conference on computer vision (ICCV)
Nister D (2004) An efficient solution to the five-point relative pose problem. IEEE Trans Pattern Anal Mach Intell 26(6):756–770
Rusinkiewicz S, Levoy M (2001) Efficient variants of the icp algorithm. In: 3DIM, pp 145–152
Snavely N, Seitz S, Szeliski R (2008) Modeling the world from internet photo collections. Int J Comput Vis 80(2):189–210
Triggs B, Mclauchlan P, Hartley R, Fitzgibbon A (2000) Bundle adjustment – a modern synthesis. In: Vision algorithms: theory and practice. LNCS, pp 298–372. Springer, Heidelberg
Tron R, Vidal R (2011) Distributed computer vision algorithms through distributed averaging. In:Â CVPR
Umeyama S (1991) Least-squares estimation of transformation parameters between two point patterns. IEEE Trans Pattern Anal Mach Intell 13(4):376–380
Varadarajan V (1984) Lie groups, lie algebras, and their representations. Springer, New York
Wilson K, Snavely N (2014) Robust global translations with 1dsfm. In: European conference on computer vision (ECCV), pp 61–75
Wu C (2013) Towards linear-time incremental structure from motion. In: Proceedings of the international conference on 3D vision, 3DV ‘13, pp 127–134
Zach C, Klopschitz M, Pollefeys M (2010) Disambiguating visual relations using loop constraints. In:Â CVPR
Zacur E, Bossa M, Olmos S (2014) Left-invariant riemannian geodesics on spatial transformation groups. SIAM J Imaging Sci 7(3):1503–1557
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The author thanks an anonymous reviewer and Avishek Chatterjee for critical and useful comments.
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Govindu, V.M. (2016). Motion Averaging in 3D Reconstruction Problems. In: Turaga, P., Srivastava, A. (eds) Riemannian Computing in Computer Vision. Springer, Cham. https://doi.org/10.1007/978-3-319-22957-7_7
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DOI: https://doi.org/10.1007/978-3-319-22957-7_7
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