Abstract
In this paper, the effect of the existence of a critical set for the projective reconstruction of a scene in \({{\mathbb {P}}}^{3}\) from two views is analyzed directly on the image planes. Corresponding points, which are images of critical points, are linked by a birational map between the two planes which is a quadratic transformation. This transformation is explicitly described and used to investigate the instability phenomena for reconstruction with a new approach.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beltrametti, M., Carletti, E., Gallarati, D.: Lectures on Curves. Surfaces and Projective Varieties: A Classical View of Algebraic Geometry. EMS Textbooks in Mathematics, vol. 9. European Mathematical Society, Zürich (2009)
Bertolini, M., Besana, G.M., Turrini, C.: Instability of projective reconstruction from 1-view near critical configurations in higher dimensions. In: Alberto, C., Juan, M., Claudia, P. (eds.) Algebra Geometry and their Interactions. Contemporary Mathematics, vol. 448, pp. 1–12 (2007)
Bertolini, M., Besana, G.M., Turrini, C.: Instability of projective reconstruction of dynamic scenes near critical configurations. In: Proceedings of the International Conference on Computer Vision, ICCV (2007)
Bertolini, M., Besana, G.M., Turrini, C.: Applications of multiview tensors in higher dimensions. In: Aja-Fernández, S., de Luis Garcia, R., Tao, D., Li, X. (eds.) Tensors in Image Processing and Computer Vision. Advances in Pattern Recognition. Springer, Berlin (2009)
Bertolini, M., Besana, G.M., Turrini, C.: Critical loci for projective reconstruction from multiple views in higher dimension: a comprehensive theoretical approach. Linear Algebra Appl. 469, 335–363 (2015)
Bertolini, M., Notari, R., Turrini, C.: The Bordiga surface as critical locus for 3-view reconstructions. J. Symb. Comput. 91, 74–97 (2017). (Special Issue MEGA)
Bertolini, M., Turrini, C.: Critical configurations for 1-view in projections from \({\mathbb{P}}^{k} \rightarrow {\mathbb{P}}^{2}\). J. Math. Imaging Vis. 27, 277–287 (2007)
Buchanan, T.: The twisted cubic and camera calibration. Comput. Vis. Graph. Image Process. 42, 130–132 (1988)
Dolgachev, I.V.: Classical Algebraic Geometry: A Modern View. Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/CBO9781139084437
Faugeras, O.: Three-Dimensional Computer Vision. A Geometric Viewpoint. MIT Press, Cambridge (1993)
Hartley, R., Kahl, F.: Critical configurations for projective reconstruction from multiple views. Int. J. Comput. Vis. 71(1), 5–47 (2007)
Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2003). (With a foreword by Olivier Faugeras)
Hudson, H.P.: Cremona Transformation in Plane and Space. Cambridge University Press, Cambridge (1927)
Krames, J.: Zur ermittlung eines objectes aus zwei perspectiven (ein beitrag zur theorie der gefhrlichen rter). Monatsh. Math. Phys. 49, 327–354 (1940)
Maybank, S.: Theory of Reconstruction from Image Motion. Springer, Secaucus (1992)
Semple, J.G., Kneebone, G.T.: Algebraic Projective Geometry. Oxford University Press, Oxford (1952)
Sturm, P.: Critical motion sequences for the self-calibration of cameras and stereo systems with variable focal length. Image Vis. Comput. 20, 415–426 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bertolini, M., Magri, L. & Turrini, C. Critical Loci for Two Views Reconstruction as Quadratic Transformations Between Images. J Math Imaging Vis 61, 1322–1328 (2019). https://doi.org/10.1007/s10851-019-00908-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-019-00908-w