The browser you are using is not supported by this website. All versions of Internet Explorer are no longer supported, either by us or Microsoft (read more here: https://www.microsoft.com/en-us/microsoft-365/windows/end-of-ie-support).

Please use a modern browser to fully experience our website, such as the newest versions of Edge, Chrome, Firefox or Safari etc.

Algebraic Varieties in Multiple View Geometry

På svenska

Author

Editor

  • R. Cipolla
  • B. Buxton

Summary, in English

In this paper we will investigate the different algebraic varieties and ideals that can be generated from multiple view geometry with uncalibrated cameras. The natural descriptor, Vn, is the image of p3 in P2×p2×...×p2 under n different projections. However, we will show that Vn is not a variety. Another descriptor, the variety Vb, is generated by all bilinear forms between pairs of views and consists of all points in p2×p2×...×p2 where all bilinear forms vanish. Yet another descriptor, the variety, Vt, is the variety generated by all trilinear forms between triplets of views. We will show that when n=3, Vt is a reducible variety with one component corresponding to Vb and another corresponding to the trifocal plane. In ideal theoretic terms this is called a primary decomposition. This settles the discussion on the connection between the bilinearities and the trilinearities. Furthermore, we will show that when n=3, Vt is generated by the three bilinearities and one trilinearity and when n⩾4, Vt is generated by the (2n) bilinearities. This shows that four images is the generic case in the algebraic setting, because Vt can be generated by just bilinearities

Publishing year

1996

Language

English

Pages

671-682

Publication/Series

Computer Vision - ECCV '96. 4th Eurpean Conference on Computer Proceedings

Volume

2

Document type

Conference paper

Publisher

Springer

Topic

  • Mathematics

Keywords

  • algebra
  • computer vision
  • image sequences
  • polynomials
  • multiple view geometry
  • uncalibrated cameras
  • natural descriptor
  • bilinear forms
  • trilinear forms
  • triplets of views
  • trifocal plane
  • ideal theoretic terms
  • primary decomposition
  • algebraic setting
  • polynomial equations

Conference name

Proceedings of Fourth European Conference on Computer Vision. ECCV '96

Conference date

1996-04-14 - 1996-04-18

Conference place

Cambridge, United Kingdom

Status

Published

ISBN/ISSN/Other

  • ISBN: 3 540 61123 1