General Techniques for Approximate Incidences and Their Application to the Camera Posing Problem

Authors Dror Aiger, Haim Kaplan, Efi Kokiopoulou, Micha Sharir, Bernhard Zeisl

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Dror Aiger
  • Google, Tel Aviv, Israel
Haim Kaplan
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel
  • Google, Tel Aviv, Israel
Efi Kokiopoulou
  • Google, Zurich, Switzerland
Micha Sharir
  • School of Computer Science, Tel Aviv University, Tel Aviv, Israel
Bernhard Zeisl
  • Google, Zurich, Switzerland

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Dror Aiger, Haim Kaplan, Efi Kokiopoulou, Micha Sharir, and Bernhard Zeisl. General Techniques for Approximate Incidences and Their Application to the Camera Posing Problem. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We consider the classical camera pose estimation problem that arises in many computer vision applications, in which we are given n 2D-3D correspondences between points in the scene and points in the camera image (some of which are incorrect associations), and where we aim to determine the camera pose (the position and orientation of the camera in the scene) from this data. We demonstrate that this posing problem can be reduced to the problem of computing epsilon-approximate incidences between two-dimensional surfaces (derived from the input correspondences) and points (on a grid) in a four-dimensional pose space. Similar reductions can be applied to other camera pose problems, as well as to similar problems in related application areas. We describe and analyze three techniques for solving the resulting epsilon-approximate incidences problem in the context of our camera posing application. The first is a straightforward assignment of surfaces to the cells of a grid (of side-length epsilon) that they intersect. The second is a variant of a primal-dual technique, recently introduced by a subset of the authors [Aiger et al., 2017] for different (and simpler) applications. The third is a non-trivial generalization of a data structure Fonseca and Mount [Da Fonseca and Mount, 2010], originally designed for the case of hyperplanes. We present and analyze this technique in full generality, and then apply it to the camera posing problem at hand. We compare our methods experimentally on real and synthetic data. Our experiments show that for the typical values of n and epsilon, the primal-dual method is the fastest, also in practice.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Camera positioning
  • Approximate incidences
  • Incidences


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  1. Sameer Agarwal, Noah Snavely, Ian Simon, Steven M Seitz, and Richard Szeliski. Building Rome in a day. In Commun. ACM 54(10), pages 105-112. ACM, 2011. Google Scholar
  2. D. Aiger, H. Kaplan, E. Kokiopoulou, M. Sharis, and B. Zeisl. General techniques for approximate incidences and their application to the camera posing problem. CoRR, 2019. URL:
  3. Dror Aiger, Haim Kaplan, and Micha Sharir. Output Sensitive Algorithms for Approximate Incidences and Their Applications. In Computational Geometry, to appear. Also in European Symposium on Algorithms, volume 5, pages 1-13, 2017. Google Scholar
  4. Guilherme D Da Fonseca and David M Mount. Approximate range searching: The absolute model. Computational Geometry, 43(4):434-444, 2010. Google Scholar
  5. Martin A Fischler and Robert C Bolles. Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography. Communications of the ACM, 24(6):381-395, 1981. Google Scholar
  6. Christian Häne, Lionel Heng, Gim Hee Lee, Friedrich Fraundorfer, Paul Furgale, Torsten Sattler, and Marc Pollefeys. 3D visual perception for self-driving cars using a multi-camera system: Calibration, mapping, localization, and obstacle detection. Image and Vision Computing, 68:14-27, 2017. Google Scholar
  7. Bert M Haralick, Chung-Nan Lee, Karsten Ottenberg, and Michael Nölle. Review and analysis of solutions of the three point perspective pose estimation problem. International Journal of Computer Vision, 13(3):331-356, 1994. Google Scholar
  8. Georg Klein and David Murray. Parallel Tracking and Mapping for Small AR Workspaces. In ISMAR, pages 83-86. IEEE, 2009. Google Scholar
  9. Sven Middelberg, Torsten Sattler, Ole Untzelmann, and Leif Kobbelt. Scalable 6-DOF Localization on Mobile Devices. In European Conference on Computer Vision, pages 268-283. Springer, 2014. Google Scholar
  10. Marc Pollefeys, Luc Van Gool, Maarten Vergauwen, Frank Verbiest, Kurt Cornelis, Jan Tops, and Reinhard Koch. Visual Modeling with a Hand-Held Camera. International Journal of Computer Vision, 59(3):207-232, 2004. Google Scholar
  11. Johannes L Schonberger and Jan-Michael Frahm. Structure-from-Motion Revisited. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4104-4113, 2016. Google Scholar
  12. Chris Sweeney, John Flynn, Benjamin Nuernberger, Matthew Turk, and Tobias Höllerer. Efficient Computation of Absolute Pose for Gravity-Aware Augmented Reality. In ISMAR, pages 19-24. IEEE, 2015. Google Scholar
  13. Bernhard Zeisl, Torsten Sattler, and Marc Pollefeys. Camera Pose Voting for Large-Scale Image-Based Localization. In Proceedings of the IEEE International Conference on Computer Vision, pages 2704-2712, 2015. Google Scholar