Skip to main content
SpringerLink
Log in

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

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alziary de Roquefort, B. (1991): Jeux différentiels et approximation de fonction valeur. RAIRO (M 2 AN)25, n. 5, 535–560

    Google Scholar 

  2. Barles, G. (1990): An approach of deterministic control problems with unbounded data. Ann. Inst. Henri Poincaré7, n. 4, 235–258

    Google Scholar 

  3. Barles, G., Perthame, B. (1987): Discontinuous solutions of deterministic optimal stopping time problems. Math. Mod. Anal. Numer.21, n. 4, 557–579

    Google Scholar 

  4. Barles, G., Perthame, B. (1988): Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optimization26, 1133–1148

    Google Scholar 

  5. Barles, G., Souganidis, P.E. (1991) Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal.4, 271–283

    Google Scholar 

  6. Capuzzo-Dolcetta, I. (1983): On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optimization10, 367–377

    Google Scholar 

  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.), 161–181

    Google Scholar 

  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, 487–502

    Google Scholar 

  9. Crandall, M.G., Ishii, H., Lions, P.L. (1987): Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited. J. Math. Soc. Japan39, n. 4

    Google Scholar 

  10. Crandall, M.G., Lions, P.L. (1983): Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277, 1–42

    Google Scholar 

  11. Crandall, M.G., Lions, P.L. (1984): Two approximations of solutions of Hamilton-Jacobi equations. Math. Comput.43, 167, 1–19

    Google Scholar 

  12. Falcone, M. (1985): Numerical solution of deterministic continuous control problems. Proc. Int. Symp. Numer. Anal. Madrid

  13. Falcone, M. (1987): A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optimization15, 1–13

    Google Scholar 

  14. Horn, B.K.P. Robot Vision. The MIT Engineering and Computer Science Series. The MIT Press, McGraw-Hill, New York

  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, 247–251

    Google Scholar 

  16. Ishii, H. (1985): Hamilton-Jacobi Equations with discontinuous Hamiltonians on arbitrary open subsets. Bull. Fac. Sci. Engrg. Chuo Univ.28, 33–77

    Google Scholar 

  17. Lions P.L., (1982): Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London

    Google Scholar 

  18. Osher, S.J., Rudin, L. (1992): Rapid convergence of approximate solutions to shape-from-shading problems. To appear

  19. Pentland, A.P. (1984): Local analysis of the image. IEEE Trans. Pattern Anal. Mach. Recog.6(2), 170–187

    Google Scholar 

  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. Pentland, A.P. (1990): Linear shape-from-shading. Int. J. Comput. Vision4, 153–162

    Google Scholar 

  22. Souganidis, P.E. (1985): Approximation schemes for viscosity solutions of Hamilton-Jacobi equations. J. Differ. Equations59, 1–43

    Google Scholar 

  23. Rouy, E., Tourin, A. (1992): A viscosity solution approach to Shape-from-Shading. Siam J. Numer. Anal. (to appear)

Download references

Author information

Authors and Affiliations

  1. CEREMADE, Université Paris IX-Dauphine, Place de Lattre de Tassigny, F-75775, Paris Cédex 16, France

    Agnès Tourin

Authors
  1. Agnès Tourin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tourin, A. A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and application to Shape-from-Shading problems. Numer. Math. 62, 75–85 (1992). https://doi.org/10.1007/BF01396221

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01396221

Mathematics Subject Classification (1991)

Search

Navigation