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Foundations of Computational Mathematics

, Volume 18, Issue 4, pp 1043–1071 | Cite as

Distortion Varieties

  • Joe Kileel
  • Zuzana Kukelova
  • Tomas Pajdla
  • Bernd Sturmfels
Article

Abstract

The distortion varieties of a given projective variety are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions. These are based on Chow polytopes and Gröbner bases. Multi-parameter distortions are studied using tropical geometry. The motivation for distortion varieties comes from multi-view geometry in computer vision. Our theory furnishes a new framework for formulating and solving minimal problems for camera models with image distortion.

Keywords

Distortion varieties Toric varieties Image distortion Minimal problems 

Mathematics Subject Classification

14Q15 13P15 68T45 14M25 13P25 14T05 68U10 

Notes

Acknowledgements

This project started at the Algebraic Vision workshop (May 2016) at the American Institute of Mathematics (AIM) in San Jose. We are grateful to the organizers, Sameer Agarwal, Max Lieblich and Rekha Thomas, for bringing us together. Joe Kileel and Bernd Sturmfels were supported by the US National Science Foundation (DMS-1419018). Zuzana Kukelova was supported by the Czech Science Foundation (GACR P103/12/G084). Part of this study was carried out while she worked for Microsoft Research, Cambridge, UK. Tomas Pajdla was supported by H2020-ICT-731970 LADIO, the Robotics for Industry 4.0 Project CZ.02.1.01/0.0/0.0/15_003/0000470 and the European Regional Development Fund.

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Copyright information

© SFoCM 2017

Authors and Affiliations

  • Joe Kileel
    • 1
  • Zuzana Kukelova
    • 2
  • Tomas Pajdla
    • 3
  • Bernd Sturmfels
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of Cybernetics, Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic
  3. 3.Czech Institute of Informatics, Robotics and CyberneticsCzech Technical University in PraguePragueCzech Republic

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