Advertisement

Shape from Chebyshev nets

  • Jan Koenderink
  • Andrea van Doom
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1407)

Abstract

We consider a special type of wiremesh covering arbitrarily curved (but smooth) surfaces that conserves length in two distinct directions at every point of the surface. Such “Chebyshev nets” can be considered as deformations of planar Cartesian nets (chess boards) that conserve edge lengths but sacrifice orthogonality of the parameter curves. A unique Chebyshev net can be constructed when two intersecting parameter curves are arbitrarily specified at a point of the surface. Since any Chebyshev net can be applied to the plane, such nets induce mappings between any arbitrary pair of surfaces. Such mappings have many desirable properties (much freedom, yet conservation of length in two directions). Because Chebyshev nets conserve edge lengths they yield very strong constraints on the projection. As a result one may compute the shape of the surface from a single view if the assumption that one looks at the projection of a Chebyshev net holds true. The structure of the solution is a curious one and warrants attention. Human observers apparently are able to use such an inference witness the efficaciousness of fishnet stockings and body-suits in optically revealing the shape of the body. We argue that Chebyshev nets are useful in a variety of common tasks.

Keywords

Edge Length Human Observer Cartesian Grid Single View Depth Difference 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1927]
    Bianchi, L.: Lezioni di geometria differenziale. 3rd ed., Vol. I, Part I, Nicola Zarichelli, Bologna (1927) 153–162Google Scholar
  2. [1993]
    Faugeras, O.: Three-dimensional computer vision. The MIT Press, Cambridge, Mass. (1993)Google Scholar
  3. [1986]
    Koenderink, J. J.: Optic Flow. Vision Research 26 (1986) 161–180CrossRefGoogle Scholar
  4. [1970]
    Sauer, R.: Differenzengeometrie. Springer, Berlin (1970)Google Scholar
  5. [1981]
    Stevens, K. A.: The visual interpretation of surface contours. Artificial Intelligence 17 (1981) 47–73CrossRefGoogle Scholar
  6. [1983]
    Stevens, K. A.: The line of curvature constraint and the interpretation of 3-D shape from parallel surface contours. 8th Int.Joint Conf. on Artificial Intelligence (1983) 1057–1062Google Scholar
  7. [1986]
    Stevens, K. A.: Inferring shape from contours across surfaces. In: From pixels to predicates. Ed. A. P. Pentland, Ablex Publ. Corp., Norwood, NJ. (1986)Google Scholar
  8. [1942]
    Thomas, H.: Zur Frage des Gleichgewichts von Tschebyscheff-Netzen aus verknoteten und gespannten FÄden. Math.Z. 47 (1942) 66–77CrossRefGoogle Scholar
  9. [1878]
    Tschebyscheff, P. L.: Sur la coupe des vÊtements. Association francaise pour l'avancement des sciences. Congrès de Paris (1878) 154Google Scholar
  10. [1882]
    Voss, A.: über ein neues Prinzip der Abbildung krummer OberflÄchen. Math.Ann. XIX (1882) 1–25MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jan Koenderink
    • 1
  • Andrea van Doom
    • 2
  1. 1.Helmholtz InstituutUniversiteit UtrechtTA UtrechtThe Netherlands
  2. 2.Laboratory for Form Theory, Fac. of Industrial Design Eng.Technical University DelftBX DelftThe Netherlands

Personalised recommendations