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On Image Contours of Projective Shapes

  • Jean Ponce
  • Martial Hebert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8692)

Abstract

This paper revisits classical properties of the outlines of solid shapes bounded by smooth surfaces, and shows that they can be established in a purely projective setting, without appealing to Euclidean measurements such as normals or curvatures. In particular, we give new synthetic proofs of Koenderink’s famous theorem on convexities and concavities of the image contour, and of the fact that the rim turns in the same direction as the viewpoint in the tangent plane at a convex point, and in the opposite direction at a hyperbolic point. This suggests that projective geometry should not be viewed merely as an analytical device for linearizing calculations (its main role in structure from motion), but as the proper framework for studying the relation between solid shape and its perspective projections. Unlike previous work in this area, the proposed approach does not require an oriented setting, nor does it rely on any choice of coordinate system or analytical considerations.

Keywords

Tangent Plane Projective Geometry Perspective Projection Visual Hull Gaussian Image 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jean Ponce
    • 1
  • Martial Hebert
    • 2
  1. 1.Department of Computer ScienceEcole Normale SupérieureFrance
  2. 2.Robotics InstituteCarnegie-Mellon UniversityUSA

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