Well-posedness of linear shape-from-shading problem

  • Ryszard Kozera
  • Reinhard Klette
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1296)


We continue to study here a global shape recovery of a smooth surface for which the reflectance map is linear. It was recently proved that under special conditions the corresponding finite difference based algorithms are stable and thus convergent to the ideal solution. The whole analysis was based on the assumption that the problem related to the linear image irradiance equation is well-posed. Indeed, we show in this paper that under certain conditions there exists a unique global C2 solution (depending continuously on the initial data) to the corresponding Cauchy problem defined over the entire image domain (with non-smooth boundary).


Cauchy Problem Global Solution Global Existence Class Global Solution Simple Geometric Interpretation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold V. I.: Ordinary Differential Equation. MIT Press Cambridge MA (1973)Google Scholar
  2. 2.
    John F.: Partial Differential Equations. Vol. 1 Springer-Verlag New York (1971)Google Scholar
  3. 3.
    Kozera R.: An algorithm for linear shape-from-shading problem. In Proc. 6th International Conference on Computer Analysis of Images and Patterns. Springer-Verlag Berlin-Heidelberg, Prague, Czech Republic (September 1995) 408–415Google Scholar
  4. 4.
    Kozera R. and Klette R.: Finite difference based algorithms for linear shape from shading. Machine Graphics and Vision (to appear)Google Scholar
  5. 5.
    Lax P. D. and Richtmyer R. D.: Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9 (1956) 267–293Google Scholar
  6. 6.
    Maurin K.: Analysis. Vol. 1 PWN-Polish Scientific Publishers (1973)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Ryszard Kozera
    • 1
  • Reinhard Klette
    • 2
  1. 1.Department of Computer ScienceThe University of Western AustraliaNedlandsAustralia
  2. 2.Tamaki Campus, Computer Science DepartmentThe Auckland UniversityAucklandNew Zealand

Personalised recommendations