Non-visible deformations

  • Jean-Denis Durou
  • Laurent Mascarilla
  • Didier Piau
Poster Session B: Active Vision, Motion, Shape, Stereo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1310)


The number of non isomorphic solutions of the eikonal equation can be infinite. We show that there can exist a whole family of non isomorphic solutions, indexed by a continuous parameter. This implies, first, that the general problem of shape from shading can be ill-posed when no additional condition on the shape is imposed. This is in contradiction with what is sometimes stated in the literature of shape from shading. Furthermore, this implies that there can exist non visible deformations of a given surface, i. e., continuous deformations of the surface which do not modify the image.


shape from shading eikonal equation ill-posed problem shape reconstruction 


  1. 1.
    Horn B.K.P. “Shape from Shading: a Method for Obtaining the Shape of a Smooth Opaque Object from One View”. PhD thesis, Department of Electrical Engineering, MIT, 1970.Google Scholar
  2. 2.
    Horn B.K.P. “Obtaining Shape from Shading Information”. In The Psychology of Computer Vision, chapter 4, pages 115–155. P.H. Winston (ed.), New York, 1975.Google Scholar
  3. 3.
    Bruss A.R. “The Eikonal Equation: Some Results Applicable to Computer Vision”. Journal of Mathematical Physics, 23(5):890–896, 1982.Google Scholar
  4. 4.
    Oliensis J. “Uniqueness in Shape from Shading”. International Journal of Computer Vision, 6(2):75–104, 1991.Google Scholar
  5. 5.
    Brooks M.J., Chojnacki W., and Kozera R. “Shading without Shape”. Quarterly of Applied Mathematics, L(1):27–38, 1992.Google Scholar
  6. 6.
    Horn B.K.P., Szeliski R.S., and Yuille A.L. “Impossible Shaded Images”. IEEE PAMI, 15(2):166–169, 1993.Google Scholar
  7. 7.
    Blake A., Zisserman A., and Knowles G. “Surface Descriptions from Stereo and Shading”. Image and Vision Computing, 3(4):183–191, 1985.Google Scholar
  8. 8.
    Durou J.D. “Reconnaissance du relief à partir de l'éclairement”. PhD thesis, Université Paris XI-Orsay, 1993.Google Scholar
  9. 9.
    Durou J.D., Mascarilla L., and Piau D. “Non visible distortions”. Submitted to IEEE PAMI.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Jean-Denis Durou
    • 1
  • Laurent Mascarilla
    • 1
  • Didier Piau
    • 2
  1. 1.Université Paul Sabatier (Toulouse III) IRIT 118Toulouse CedexFrance
  2. 2.Université Claude Bernard (Lyon I) Laboratoire de Probabilités 43Villeurbanne CedexFrance

Personalised recommendations