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General Deformations of Point Configurations Viewed By a Pinhole Model Camera

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Abstract

This paper is a theoretical study of the non-rigid structure from motion problem: what can be computed from a monocular view of a parametrically deforming set of points? We treat various variations of this problem for 3D affine and general smooth deformations (under some mild technical restrictions) with either a calibrated or an uncalibrated camera. We show that in general at least three images related by quasi-identical deformations are needed to have a finite set of solutions to the points’ structure.

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References

  1. Akhter, I., Khan, S., Sheikh, Y., Kanade, T.: Nonrigid structure from motion in trajectory space, Adv. Neur. Inform. Proc. Syst. (2008)

  2. Shashua, A., Avidan, S., Werman, M.: Trajectory triangulation over conic sections. In: The Proceedings of the Seventh IEEE International Conference on Computer Vision, ICCV (1999)

  3. Angst, R., Pollefeys, M.: A Unified View on Deformable Shape Factorizations, ECCV (2012)

  4. Dai, Y., Li, H., He, Mingyi: A simple prior-free method for non-rigid structure-from-motion factorization, CVPR (2012)

  5. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  6. Gotardo, P., Martínez, A.: Computing smooth time trajectories for camera and deformable shape in structure from motion with occlusion. PAMI 33, 10 (2011)

    Article  Google Scholar 

  7. Greuel, G.M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  8. Harris, J.: Algebraic Geometry, a First Course. Springer, Berlin (1992)

    MATH  Google Scholar 

  9. Hartley, R.: Defense of the eight-point algorithm, In: Transactions of Pattern Analysis and Machine Intelligence, pp. 580–593 (1997)

  10. Hartley, R., Vidal, R.: Perspective nonrigid shape and motion recovery, ECCV (2008)

  11. Hartley, R., Zisserman, A.: Multiple-View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  12. Jamalifar, H., Ghadakchi, V., Kasaei, S.: Reference-free monocular 3D tracking of deformable surfaces, ISSPA (2012)

  13. Yan, J., Pollefeys, M.: A Factorization-Based Approach for Articulated Nonrigid Shape, Motion and Kinematic Chain Recovery From Video, PAMI, 30 (2008)

  14. Kaminski, J.Y., Teicher, M.: A general framework for trajectory triangulation. J. Mathe. Imaging Vis. 21(1), 27–41 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kaminski, J.Y., Teicher, M.: General trajectory triangulation. In: European Conference of Computer Vision. pp. 823–836 (2002)

  16. Kaminski, J.Y.: Algebraic Curves in Multiple-View Geometry: An algebraic geometry approach to computer vision. LAP LAMBERT Academic Publishing, Saarbruecken (2011)

    Google Scholar 

  17. Lee, J.: Introduction to Smooth Manifolds, 2nd edn. Springer, Berlin (2013)

    MATH  Google Scholar 

  18. Levin, A., Wolf, L., Shashua, A.: Time-varying Shape Tensors for Scenes with Multiply Moving Points. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2001)

  19. Mather, J.: Notes on topological stability. Bull. Am. Math. Soc. 49, 475–506 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Montagnat, J., Delingette, H., Ayache, N.: A review of deformable surfaces: topology, geometry and deformation. Image Vis. Comput. 19, 1023–1040 (2001)

    Article  Google Scholar 

  21. Salzmann, M.: Learning and Recovering 3D Surface Deformations EPFL Thesis N 4270 (2009)

  22. Saunders, D.J.: The Geometry of Jet Bundles. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  23. Taylor, J., Jepson, A., Kutulakos, K.: Non-rigid structure from locally-rigid motion, CVPR (2010)

  24. Torr, P.: Bayesian model estimation and selection for Epipolar geometry and generic manifold fitting. Int. J. Comput. Vis. 50(1), 35–61 (2002)

    Article  MATH  Google Scholar 

  25. Vidal, R., Soatto, S., Ma, Y., Sastry, S.: Segmentation of Dynamic Scenes from the Multibody Fundamental Matrix, ECCV Workshop on Vision and Modeling of Dynamic Scenes (2002)

  26. Wolf, Lior, Shashua, A.: On Projection Matrices \(P^k \rightarrow P^2\), \(k=3,\ldots ,6\), and their applications in computer vision. In: International Conference on Computer Vision (ICCV) (2001)

  27. Xiao, J., Chai, J., Kanade, T.: A closed-form solution to non-rigid shape and motion recovery, ECCV (2004)

  28. Innmann, M., Kim, K., Gu, J., Nießner, M., Loop, C., Stamminger, M., Kautz, J.: Nrmvs: Non-rigid multi-view stereo. In: Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pp. 2754–2763 (2020)

  29. Bozic, A., Palafox, P., Zollhofer, M., Thies, J., Dai, A., Nießner, M.: Neural deformation graphs for globally-consistent non-rigid reconstruction. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 1450–1459 (2021)

  30. Deng, B., Yao, Y., Dyke, R.M., Zhang, J.: A srvey of non-rigid 3D registration. Comput. Graph. Forum 41, 559–589 (2022). https://doi.org/10.1111/cgf.14502

    Article  Google Scholar 

  31. Tang, J., Xu, D., Jia K., Zhang, L.: Learning Parallel Dense Correspondence from Spatio-Temporal Descriptors for Efficient and Robust 4D Reconstruction. CVPR (2021)

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Authors and Affiliations

  1. Department of Mathematics, Holon Institute of Technology, Golomb, 52000, Holon, Israel

    Yirmeyahu Kaminski

  2. School of Computer Sciences and Engineering, The Hebrew University of Jerusalem, Givat-Ram, Jerusalem, Israel

    Michael Werman

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  1. Yirmeyahu Kaminski
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  2. Michael Werman
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Correspondence to Yirmeyahu Kaminski.

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Kaminski, Y., Werman, M. General Deformations of Point Configurations Viewed By a Pinhole Model Camera. J Math Imaging Vis 65, 631–643 (2023). https://doi.org/10.1007/s10851-023-01142-1

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