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Condition numbers in multiview geometry, instability in relative pose estimation, and RANSAC (2310.02719v1)

Published 4 Oct 2023 in cs.CV, cs.NA, and math.NA

Abstract: In this paper we introduce a general framework for analyzing the numerical conditioning of minimal problems in multiple view geometry, using tools from computational algebra and Riemannian geometry. Special motivation comes from the fact that relative pose estimation, based on standard 5-point or 7-point Random Sample Consensus (RANSAC) algorithms, can fail even when no outliers are present and there is enough data to support a hypothesis. We argue that these cases arise due to the intrinsic instability of the 5- and 7-point minimal problems. We apply our framework to characterize the instabilities, both in terms of the world scenes that lead to infinite condition number, and directly in terms of ill-conditioned image data. The approach produces computational tests for assessing the condition number before solving the minimal problem. Lastly synthetic and real data experiments suggest that RANSAC serves not only to remove outliers, but also to select for well-conditioned image data, as predicted by our theory.

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References (50)
  1. D. Ablan. Digital Photography for 3D Imaging and Animation. Wiley, 2007.
  2. Building Rome in a day. Communications of the ACM, 54(10):105–112, 2011.
  3. Learning to find good models in ransac. In CVPR, pages 15744–15753, 2022.
  4. MAGSAC++, a fast, reliable and accurate robust estimator. In CVPR, pages 1304–1312, 2020.
  5. Machine learning the real discriminant locus. Journal of Symbolic Computation, 115:409–426, 2023.
  6. N. Bourbaki. General Topology: Chapters 1–4, volume 18. Springer, 2013.
  7. E. Brachmann and C. Rother. Neural-guided RANSAC: Learning where to sample model hypotheses. In CVPR, pages 4322–4331, 2019.
  8. M. Bråtelund. Critical configurations for two projective views, a new approach. Journal of Symbolic Computation, 120:102226, 2024.
  9. P. Breiding and S. Timme. HomotopyContinuation. jl: A package for homotopy continuation in Julia. In Mathematical Software–ICMS 2018: 6th International Conference, South Bend, IN, USA, July 24-27, 2018, Proceedings 6, pages 458–465. Springer, 2018.
  10. P. Breiding and N. Vannieuwenhoven. The condition number of Riemannian approximation problems. SIAM Journal on Optimization, 31(1):1049–1077, 2021.
  11. P. Burgisser. Condition of intersecting a projective variety with a varying linear subspace. SIAM Journal on Applied Algebra and Geometry, 1(1):111–125, 2017.
  12. P. Bürgisser and F. Cucker. Condition: The Geometry of Numerical Algorithms, volume 349. Springer, 2013.
  13. GPU-based homotopy continuation for minimal problems in computer vision. In CVPR, pages 15765–15776, 2022.
  14. Parallel path tracking for homotopy continuation using GPU. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, 2022.
  15. Locally optimized RANSAC. In Joint Pattern Recognition Symposium, pages 236–243. Springer, 2003.
  16. The geometry of rank drop in a class of face-splitting matrix products. arXiv preprint arXiv:2301.09826, 2023.
  17. M. Demazure. Sur Deux Problemes De Reconstruction. PhD thesis, INRIA, 1988.
  18. J. W. Demmel. On condition numbers and the distance to the nearest ill-posed problem. Numerische Mathematik, 51(3):251–289, 1987.
  19. T. Dobbert. Matchmoving: The Invisible Art of Camera Tracking. Sybex, 2005.
  20. PL1P-Point-Line minimal problems under partial visibility in three views. In ECCV, pages 175–192, 2020.
  21. On the instability of relative pose estimation and RANSAC’s role. In CVPR, pages 8935–8943, 2022.
  22. The Chow form of the essential variety in computer vision. Journal of Symbolic Computation, 86:97–119, 2018.
  23. Matrix Computations. JHU press, 2013.
  24. Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/.
  25. J. Harris. Algebraic Geometry: A First Course, volume 133. Springer, 2013.
  26. R. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. CUP, 2nd edition, 2004.
  27. B. Horn. Robot Vision. MIT press, 1986.
  28. F. Kahl and R. Hartley. Critical curves and surfaces for Euclidean reconstruction. In ECCV, pages 447–462, 2002.
  29. Critical configurations for n𝑛nitalic_n-view projective reconstruction. In CVPR, volume 2, 2001.
  30. Algebraic characterization of essential matrices and their averaging in multiview settings. In CVPR, pages 5895–5903, 2019.
  31. M. Kitagawa and B. Windsor. MoCap for Artists: Workflow and Techniques for Motion Capture. Focal Press, 2008.
  32. J. Krames. Zur ermittlung eines objektes aus zwei perspektiven (ein beitrag zur theorie der “gefährlichen örter”). Monatshefte für Mathematik und Physik, 49(1):327–354, 1941.
  33. J. M. Lee. Riemannian Manifolds: An Introduction to Curvature, volume 176. Springer, 2006.
  34. J. M. Lee. Smooth Manifolds. Springer, 2013.
  35. Close Range Photogrammetry: Principles, Methods, and Applications. Wiley, 2007.
  36. S. Maybank. Theory of Reconstruction From Image Motion, volume 28. Springer, 2012.
  37. D. Mishkin. Benchmarking robust estimation methods. CVPR Tutorial: RANSAC in 2020. https://github.com/ducha-aiki/ransac-tutorial-2020-data.
  38. D. Nistér. An efficient solution to the five-point relative pose problem. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(6):756–770, 2004.
  39. A survey of structure from motion. Acta Numerica, 26:305–364, 2017.
  40. R. Penrose. A generalized inverse for matrices. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 51, pages 406–413. CUP, 1955.
  41. Image-based 3D acquisition of archaeological heritage and applications. In Conference on Virtual Reality, Archaeology, and Cultural Heritage, pages 255–262, 2001.
  42. A comparative analysis of RANSAC techniques leading to adaptive real-time random sample consensus. In ECCV, pages 500–513, 2008.
  43. Structure-from-motion revisited. In CVPR, pages 4104–4113, 2016.
  44. A comparison and evaluation of multi-view stereo reconstruction algorithms. In CVPR, pages 519–528, 2006.
  45. C. V. Stewart. Robust parameter estimation in computer vision. SIAM Review, 41(3):513–537, 1999.
  46. B. Sturmfels. The Hurwitz form of a projective variety. Journal of Symbolic Computation, 79:186–196, 2017.
  47. R. Szeliski. Computer Vision: Algorithms and Applications. Springer, 2010.
  48. Robust detection of degenerate configurations while estimating the fundamental matrix. Computer Vision and Image Understanding, 71(3):312–333, 1998.
  49. R. Tron and K. Daniilidis. The space of essential matrices as a Riemannian quotient manifold. SIAM Journal on Imaging Sciences, 10(3):1416–1445, 2017.
  50. J. Verschelde and X. Yu. Polynomial homotopy continuation on GPUs. ACM Communications in Computer Algebra, 49(4):130–133, 2016.
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