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Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 1–29 | Cite as

Changing Views on Curves and Surfaces

  • Kathlén Kohn
  • Bernd Sturmfels
  • Matthew Trager
Article
  • 169 Downloads

Abstract

Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface and observe that these coincide with the iterated singular loci of the coisotropic hypersurfaces associated with the original curve or surface. We derive formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and show how to compute exact representations for all visual event surfaces using algebraic methods.

Keywords

Computer vision Projections Contour curve Enumerative geometry 

Mathematics Subject Classification (2010)

Primary 14Q10 65D19; Secondary 68W30 53A05 

Notes

Acknowledgements

This project started in May 2016 at the GOAL workshop in Paris. We thank Mohab Safey El Din, Jean-Charles Faugère, Jon Hauenstein, and Jean Ponce for their help in the initial stages. We are also grateful to Joachim Rieger for an inspiring discussion on singularity theory, and to Emre Sertöz for helpful comments on intersection theory. Kathlén Kohn was funded by the Einstein Foundation Berlin. Bernd Sturmfels received partial support from the US National Science Foundation (DMS-1419018) and the Einstein Foundation Berlin. Matthew Trager was supported in part by the ERC advanced grant VideoWorld, the Institut Universitaire de France, the Inria-CMU associated team GAYA, and the ANR grant RECAP.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.TU BerlinBerlinGermany
  2. 2.MPI LeipzigLeipzigGermany
  3. 3.UC BerkeleyBerkeleyUSA
  4. 4.Inria, École Normale Supérieure Paris, CNRSPSL Research UniversityParisFrance

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