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Abstrakty
In the context of multiple view geometry, images of static scenes are modeled as linear projections from a projective space ℙ3 to a projective plane ℙ2 and, similarly, videos or images of suitable dynamic or segmented scenes can be modeled as linear projections from ℙk to ℙh, with k > h ≥ 2. In those settings, the projective reconstruction of a scene consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a configuration of points and of centers of projections, in the ambient space, where the reconstruction of a scene fails. Critical loci turn out to be suitable algebraic varieties. In this paper we investigate those critical loci which are hypersurfaces in high dimension complex projective spaces, and we determine their equations. Moreover, to give evidence of some practical implications of the existence of these critical loci, we perform a simulated experiment to test the instability phenomena for the reconstruction of a scene, near a critical hypersurface.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
3--20
Opis fizyczny
Bibliogr. 27 poz., wykr.
Twórcy
autor
- Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italy
autor
- Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB), Politecnico di Milano, Milano, Italy
Bibliografia
- [1] M. Bertolini, G. Besana, R. Notari, and C. Turrini. Critical loci in computer vision and matrices dropping rank in codimension one. J. of Pure and Applied Algebra, 224(12): 106439, 2020. doi:10.1016/j.jpaa.2020.106439.
- [2] M. Bertolini, G. Besana, and C. Turrini. Applications of multiview tensors in higher dimensions. In S. Aja-Fernández, R. de Luis García, D. Tao, and X. Li, editors, Tensors in image processing and computer vision, Advances in Pattern Recognition, pages 237–260. Springer, London, 2009. doi:10.1007/978-1-84882-299-311.
- [3] M. Bertolini, G. Besana, and C. Turrini. Critical loci for projective reconstruction from multiple views in higher dimension: A comprehensive theoretical approach. Linear Algebra and its Applications, 469: 335–363, 2015. doi:10.1016/j.laa.2014.11.021.
- [4] M. Bertolini, G. Besana, and C. Turrini. Generalized fundamental matrices as grassmann tensors. Annali di Matematica Pura ed Applicata (1923 -), Jul 2016. doi:10.1007/s10231-016-0585-4.
- [5] M. Bertolini, R. Notari, and C. Turrini. The bordiga surface as critical locus for 3-view reconstructions. J. of Symbolic Computation, 91: 74–97, 2019. MEGA 2017, Effective Methods in Algebraic Geometry, Nice (France), June 12-16, 2017. doi:10.1016/j.jsc.2018.06.014.
- [6] M. Bertolini and C. Turrini. Critical configurations for 1-view in projections from Pk →P2. J. of Mathematical Imaging and Vision, 27: 277–287, 2007. doi:10.1007/s10851-007-0649-6.
- [7] M. Bertolini, C. Turrini, and G. Besana. Instability of projective reconstruction of dynamic scenes near critical configurations. In IEEE Int. Conf. on Computer Vision, pages 1–7, Los Alamitos ,CA, USA, 2007. IEEE Computer Society. doi:10.1109/ICCV.2007.4409100.
- [8] T. Buchanan. The twisted cubic and camera calibration. Computer Vision, Graphics, and Image Processing, 42(1): 130–132, 1988. doi:10.1016/0734-189X(88)90146-6.
- [9] D. A. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 2015. doi:10.1007/978-3-319-16721-3.
- [10] X. Fan and R. Vidal. The space of multibody fundamental matrices: Rank, geometry and projection. In R. Vidal, A. Heyden, and Y. Ma, editors, Dynamical Vision. Proc. Int. Workshop on Dynamical Vision WDV 2005, volume 4358 of Lecture Notes in Computer Science, pages 1–17. Springer, 2007. doi:10.1007/978-3-540-70932-91.
- [11] D. R. Grayson and M. Stillman. Macaulay2, a software system for research in algebraic geometry. http://macaulay2.com.
- [12] J. Harris. Agebraic Geometry: A First Course, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1992. doi:10.1007/978-1-4757-2189-8.
- [13] R. Hartley and F. Kahl. Critical configurations for projective reconstruction from multiple views. Int. J. of Computer Vision, 71(1): 5–47, 2007. doi:10.1007/s11263-005-4796-1.
- [14] R. Hartley and R. Vidal. The multibody trifocal tensor: motion segmentation from 3 perspective views. In Proc. 2004 IEEE Computer Society Conf. on Computer Vision and Pattern Recognition CVPR 2004, volume 1, pages I–I, Washington, DC, USA, 27 Jun-2 Jul 2004. doi:10.1109/CVPR.2004.1315109.
- [15] R. Hartley and A. Zisserman. Multiple View Geometry in Computer Vision. Cambridge University Press, New York, 2nd edition, 2003. doi:10.1017/CBO9780511811685.
- [16] R. I. Hartley. Ambiguous configurations for 3-view projective reconstruction. In Computer Vision. Proc. European Conf. on Computer Vision ECCV 2000, Part I, volume 1842 of Lecture Notes in Computer Science, pages 922–935, Dublin, Ireland, 26 Jun-1 Jul 2000. Springer. Doi:10.1007/3-540-45054-860.
- [17] R. I. Hartley and F. Schaffalitzky. Reconstruction from projections using Grassmann tensors. Int. J. of Computer Vision, 83(3): 274–293, 2009. doi:10.1007/s11263-009-0225-1.
- [18] R. Hartshorne. Algebraic Geometry, volume 52 of Graduate Texts in Mathematics. Springer, New York, 1977. doi:10.1007/978-1-4757-3849-0.
- [19] K. Huang, R. Fossum, and Y. Ma. Generalized rank conditions in multiple view geometry with applications to dynamical scenes. In Computer Vision. Proc. European Conf. on Computer Vision ECCV 2002, Part II, volume 2351 of Lecture Notes in Computer Science, pages 201–216, Copenhagen, Denmark, 28-31 May 2002. Springer. Doi:10.1007/3-540-47967-814.
- [20] F. Kahl, R. Hartley, and K. Astrom. Critical configurations for n-view projective reconstruction. In Proc. IEEE Computer Society Conf. on Computer Vision and Pattern Recognition CVPR 2001, pages II–II, Kauai, HI, USA, 8-14 Dec 2001. doi:10.1109/CVPR.2001.990945.
- [21] J. Krames. Zur Ermittlung eines Objektes aus zwei Perspektiven (Ein Beitrag zur Theorie der ,,gefährlichen Örter”). Monatshefte für Mathematik und Physik, 49: 327–354, 1940. doi:10.1007/BF01707311.
- [22] S. Maybank. Theory of Reconstruction from Image Motion, volume 28 of Springer Series in Information Sciences. Springer, Berlin, Heidelberg, 1993. doi:10.1007/978-3-642-77557-4.
- [23] A. Shashua and S. J. Maybank. Degeneraten point configurations of three views: Do critical surfaces exist? Technical Report TR 96-19, Hebrew University, 1996. http://www.cs.huji.ac.il/~shashua/papers/cvpr-critical.ps.gz.
- [24] The MathWorks, Inc. Matlab. Natick, MA, USA. https://www.mathworks.com.
- [25] R. Vidal and Y. Ma. A unified algebraic approach to 2-D and 3-D motion segmentation and estimation. J. of Mathematical Imaging and Vision, 25(3): 403–421, 2006. doi:10.1007/s10851-006-8286-z.
- [26] R. Vidal, Y. Ma, S. Soatto, and S. Sastry. Two-view multibody structure from motion. Int. J. of Computer Vision, 68(1): 7–25, 2006. doi:10.1007/s11263-005-4839-7.
- [27] L. Wolf and A. Shashua. On projection matrices Pk → P2, k = 3, ...,6, and their applications in computer vision. Int. J. of Computer Vision, 48(1): 53–67, 2002. doi:10.1023/A:1014855311993.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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