Numerische Mathematik

, Volume 62, Issue 1, pp 75–85 | Cite as

A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and application to Shape-from-Shading problems

  • Agnès Tourin


The reconstruction from a shaded image of a Lambertian and not self-shadowing surface illuminated by a single distant pointwise light source may be written as a first-order Hamilton-Jacobi equation.

In this paper, we continue the investigation begun in E. Rouy and A. Tourin into the uniqueness of the solution of this equation; the approach is based on the viscosity solutions theory and the dynamic programming principle.

More precisely, we concentrate here on the uniqueness of the viscosity solution of this equation in case the measured luminous intensity reflected by the surface is discontinuous along a smooth curve. We prove a general comparison result for a piecewise Lipschitz continuous Hamiltonian and illustrate it by numerical experiments.

Mathematics Subject Classification (1991)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alziary de Roquefort, B. (1991): Jeux différentiels et approximation de fonction valeur. RAIRO (M 2 AN)25, n. 5, 535–560Google Scholar
  2. 2.
    Barles, G. (1990): An approach of deterministic control problems with unbounded data. Ann. Inst. Henri Poincaré7, n. 4, 235–258Google Scholar
  3. 3.
    Barles, G., Perthame, B. (1987): Discontinuous solutions of deterministic optimal stopping time problems. Math. Mod. Anal. Numer.21, n. 4, 557–579Google Scholar
  4. 4.
    Barles, G., Perthame, B. (1988): Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optimization26, 1133–1148Google Scholar
  5. 5.
    Barles, G., Souganidis, P.E. (1991) Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal.4, 271–283Google Scholar
  6. 6.
    Capuzzo-Dolcetta, I. (1983): On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optimization10, 367–377Google Scholar
  7. 7.
    Capuzzo-Dolcetta, I., Falcone, M. (1989): Discrete dynamic programming and viscosity solutions of the Bellman equation. Ann. Inst. Henry Poincaré Anal. Non Linéaire6, (suppl.), 161–181Google Scholar
  8. 8.
    Crandall, M.G., Ferons, L.C., Lions, P.L. (1984): Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.282, 487–502Google Scholar
  9. 9.
    Crandall, M.G., Ishii, H., Lions, P.L. (1987): Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited. J. Math. Soc. Japan39, n. 4Google Scholar
  10. 10.
    Crandall, M.G., Lions, P.L. (1983): Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277, 1–42Google Scholar
  11. 11.
    Crandall, M.G., Lions, P.L. (1984): Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput.43, 167, 1–19Google Scholar
  12. 12.
    Falcone, M. (1985): Numerical solution of deterministic continuous control problems. Proc. Int. Symp. Numer. Anal. MadridGoogle Scholar
  13. 13.
    Falcone, M. (1987): A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optimization15, 1–13Google Scholar
  14. 14.
    Horn, B.K.P. Robot Vision. The MIT Engineering and Computer Science Series. The MIT Press, McGraw-Hill, New YorkGoogle Scholar
  15. 15.
    Ishii, H. (1987): A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type. Proc. Amer. Math. Soc.100, n.2, 247–251Google Scholar
  16. 16.
    Ishii, H. (1985): Hamilton-Jacobi Equations with discontinuous Hamiltonians on arbitrary open subsets. Bull. Fac. Sci. Engrg. Chuo Univ.28, 33–77Google Scholar
  17. 17.
    Lions P.L., (1982): Generalized Solutions of Hamilton-Jacobi Equations. Pitman, LondonGoogle Scholar
  18. 18.
    Osher, S.J., Rudin, L. (1992): Rapid convergence of approximate solutions to shape-from-shading problems. To appearGoogle Scholar
  19. 19.
    Pentland, A.P. (1984): Local analysis of the image. IEEE Trans. Pattern Anal. Mach. Recog.6(2), 170–187Google Scholar
  20. 20.
    Pentland, A.P. (1988): Shape information from shading: a theory about human perception. Technical Report 103, Vision Science, MIT Media Laboratory, MIT, Cambridge, Mass.Google Scholar
  21. 21.
    Pentland, A.P. (1990): Linear shape-from-shading. Int. J. Comput. Vision4, 153–162Google Scholar
  22. 22.
    Souganidis, P.E. (1985): Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differ. Equations59, 1–43Google Scholar
  23. 23.
    Rouy, E., Tourin, A. (1992): A viscosity solution approach to Shape-from-Shading. Siam J. Numer. Anal. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Agnès Tourin
    • 1
  1. 1.CEREMADEUniversité Paris IX-DauphineParis Cédex 16France

Personalised recommendations