International Journal of Computer Vision

, Volume 121, Issue 3, pp 403–415

On the Existence of Epipolar Matrices

  • Sameer Agarwal
  • Hon-Leung Lee
  • Bernd Sturmfels
  • Rekha R. Thomas
Article

DOI: 10.1007/s11263-016-0949-7

Cite this article as:
Agarwal, S., Lee, HL., Sturmfels, B. et al. Int J Comput Vis (2017) 121: 403. doi:10.1007/s11263-016-0949-7

Abstract

This paper considers the foundational question of the existence of a fundamental (resp. essential) matrix given m point correspondences in two views. We present a complete answer for the existence of fundamental matrices for any value of m. We disprove the widely held beliefs that fundamental matrices always exist whenever \(m \le 7\). At the same time, we prove that they exist unconditionally when \(m \le 5\). Under a mild genericity condition, we show that an essential matrix always exists when \(m \le 4\). We also characterize the six and seven point configurations in two views for which all matrices satisfying the epipolar constraint have rank at most one.

Keywords

Structure from motion Epipolar geometry Algebraic geometry 

Funding information

Funder NameGrant NumberFunding Note
Directorate for Mathematical and Physical Sciences
  • DMS-1418728
  • DMS-1418728
Directorate for Mathematical and Physical Sciences
  • DMS-1419018

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sameer Agarwal
    • 1
  • Hon-Leung Lee
    • 2
  • Bernd Sturmfels
    • 3
  • Rekha R. Thomas
    • 2
  1. 1.Google Inc.SeattleUSA
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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