Algebraic Varieties in Multiple View Geometry

Research output: Chapter in Book/Report/Conference proceedingPaper in conference proceeding

7 Citations (SciVal)

Abstract

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
Original languageEnglish
Title of host publicationComputer Vision - ECCV '96. 4th Eurpean Conference on Computer Proceedings
EditorsR. Cipolla, B. Buxton
PublisherSpringer
Pages671-682
Volume2
ISBN (Print)3 540 61123 1
Publication statusPublished - 1996
EventProceedings of Fourth European Conference on Computer Vision. ECCV '96 - Cambridge, United Kingdom
Duration: 1996 Apr 141996 Apr 18

Publication series

Name
Volume2

Conference

ConferenceProceedings of Fourth European Conference on Computer Vision. ECCV '96
Country/TerritoryUnited Kingdom
CityCambridge
Period1996/04/141996/04/18

Subject classification (UKÄ)

  • 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

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