Changing Views on Curves and Surfaces

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.

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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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Correspondence to Kathlén Kohn.

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Lecture at the Annual Meeting 2017 of the Vietnam Institute for Advanced Study in Mathematics

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Kohn, K., Sturmfels, B. & Trager, M. Changing Views on Curves and Surfaces. Acta Math Vietnam 43, 1–29 (2018). https://doi.org/10.1007/s40306-017-0240-1

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Keywords

  • Computer vision
  • Projections
  • Contour curve
  • Enumerative geometry

Mathematics Subject Classification (2010)

  • Primary 14Q10
  • 65D19; Secondary 68W30
  • 53A05