Hostname: page-component-cb9f654ff-9b74x Total loading time: 0 Render date: 2025-08-24T01:37:37.081Z Has data issue: false hasContentIssue false
  • English
  • Français

Projective Reconstruction in Algebraic Vision

Part of: Artificial intelligence (68Txx) Birational geometry

Published online by Cambridge University Press:  13 November 2019

Atsushi Ito ,
Makoto Miura  and
Kazushi Ueda
Atsushi Ito
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan Email: atsushi.ito@math.nagoya-u.ac.jp
Makoto Miura
Affiliation:
Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea Email: miura@kias.re.kr
Kazushi Ueda
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan Email: kazushi@ms.u-tokyo.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

Keywords

computer visionalgebraic visionmultiview geometryprojective reconstructionHilbert scheme

MSC classification

Primary: 14E05: Rational and birational maps
Secondary: 68T45: Machine vision and scene understanding

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 592 - 609
Check for updates
Copyright
© Canadian Mathematical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

A. I. was supported by Grant-in-Aid for Scientific Research (14J01881, 17K14162). M. M. was supported by Korea Institute for Advanced Study. K. U. was partially supported by Grant-in-Aid for Scientific Research (15KT0105, 16K13743, 16H03930).

References

Arbarello, E., Cornalba, M., Griffiths, P. A., and Harris, J., Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, 267, Springer-Verlag, 1985. https://doi.org/10.1007/978-1-4757-5323-3CrossRefGoogle Scholar
Aholt, C., Sturmfels, B., and Thomas, R., A Hilbert scheme in computer vision. Canad. J. Math. 65(2013), no. 5, 961–988. https://doi.org/10.4153/CJM-2012-023-2CrossRefGoogle Scholar
Gelfand, I. M., Kapranov, M. M., and Zelevinsky, A. V., Discriminants, resultants and multidimensional determinants. Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008.Google Scholar
Hartley, R. I. and Schaffalitzky, F., Reconstruction from projections using Grassmann tensors. Int. J. Comput. Vis. 83(2009), no. 3, 274–293. https://doi.org/10.1007/s11263-009-0225-1Google Scholar
Hartley, R. and Vidal, R., Perspective nonrigid shape and motion recovery. Computer Vision – ECCV 2008, Lecture Notes in Computer Science, 5302, 2008, pp. 276–289.Google Scholar
Li, B., Images of rational maps of projective spaces. Int. Math. Res. Not. IMRN 2008 no. 13, 4190–4228. https://doi.org/10.1093/imrn/rnx003Google Scholar
Lieblich, M. and Van Meter, L., Two Hilbert schemes in computer vision. arxiv:1707.09332Google Scholar
Mukai, S., Polarized K3 surfaces of genus 18 and 20. In: Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser., 179, Cambridge Univ. Press, Cambridge, 1992, pp. 264–276. https://doi.org/10.1017/CBO9780511662652.019CrossRefGoogle Scholar
Nasihatkon, B., Hartley, R., and Trumpf, J., On projective reconstruction in arbitrary dimensions. 2014 IEEE Conference on Computer Vision and Pattern Recognition (June 2014, pp. 477–484).10.1109/CVPR.2014.68Google Scholar
Ottaviani, G., Varietà proiettive di codimensione piccola. Aracne, 1995.Google Scholar
Wolf, L. and Shashua, A., On projection matrices 𝓟k to 𝓟2, k = 3, …, 6, and their applications in computer vision. Int. J. Comput. Vis. 48(2002), no. 1, 53–67. https://doi.org/10.1023/A:1014855311993CrossRefGoogle Scholar