A scheme for the shape-from-shading model with “black shadows”

  • M. Falcone
  • M. Sagona
  • A. Seghini


We propose an approximation scheme for the shape-from-shading (SFS) model which can handle “black shadows” in the image. The approach is global and does not require additional boundary conditions on the interfaces between light and black shadow areas. We analyze properties of the fully discrete scheme and show that the algorithm converges, under appropriate assumptions, to a unique solution by a fixed-point iteration. We present and discuss numerical tests.


Input Image Approximation Scheme Viscosity Solution Maximal Solution Eikonal Equation 
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  1. [1]
    Bardi, M., Capuzzo-Dolcetta, I. (1997): Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser Boston, Boston, MAMATHCrossRefGoogle Scholar
  2. [2]
    Camilli, E, Falcone, M. (1996): An approximation scheme for the maximal solution of the shape-from-shading model. In: International conference on image processing, 1996. Proceedings. Vol. I. IEEE, Piscataway, NJ, pp. 49–52Google Scholar
  3. [3]
    Camilli, F, Grüne, L. (2000): Numerical approximation of the maximal solutions for a class of degenerate Hamilton-Jacobi equations. SIAM J. Numer. Anal. 38, 1540–1560MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Camilli, R, Siconolfi, A. (1999): Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems. Indiana Univ. Math. J. 48, 1111–1131MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Daniel, P (2000): Pent-on extraire le relief d'une seul image? Thèse de doctorat. Université Paul Sabatier, ToulouseGoogle Scholar
  6. [6]
    Descombes, X., Durou, J.-D., Petit, D. (2001): Recuit simulé pour le “shape-from-shading”. In: Actes du 18e Colloque GRETSI'01. Vols. 1,2. [CNES] Toulouse, pp. 513–516Google Scholar
  7. [7]
    Durou, J.-D., Falcone, M., Sagona, M.: A survey on numerical methods for shape-from-shading, in preparationGoogle Scholar
  8. [8]
    Falcone, M. (1994): The minimum time problem and its applications to front propagation. In: Visintin, A., Buttazzo, G. (eds.): Motion by mean curvature and related topics. De Gruyter, Berhn, pp. 70–88Google Scholar
  9. [9]
    Falcone, M. (1997): Numerical solution of dynamic programming equations. In [1], pp. 472–504Google Scholar
  10. [10]
    Falcone, M., Sagona, M. (1997): An algorithm for the global solution of the shape-from-shading model. In: del Bimbo, A. (ed.): Image analysis and processing. (Lecture Notes in Computer Sciences, vol. 1310). Springer, Berlin, pp. 596–603CrossRefGoogle Scholar
  11. [11]
    Falcone, M., Makridakis, Ch. (eds.) (2001): Numerical methods for viscosity solutions and applications. World Scientific, SingaporeMATHGoogle Scholar
  12. [12]
    Horn, B.K.P., Brooks, M.J. (1986): The variational approach to shape-from-shading. Comput. Vision, Graphics Image Process. 33, 174–208MATHCrossRefGoogle Scholar
  13. [13]
    Horn, B.K.R, Brooks, M.J. (eds.) (1989): Shape-from-shading. MIT Press, Cambridge, MAGoogle Scholar
  14. [14]
    Ishii, H., Ramaswamy, M. (1995): Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients. Comm. Partial Differential Equations 20, 2187–2213MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Kimmel, R., Bruckstein, A.M. (1993): Global shape-from-shading. Center for Intelligent System Report CIS #9327, Technion — Israel Inst, of Tech., IsraelGoogle Scholar
  16. [16]
    Lions, P.-L. (1982): Generalized solutions of Hamilton-Jacobi equations. Pitman, LondonMATHGoogle Scholar
  17. [17]
    Lions, P.-L., Rouy, E., Tourin, A. (1993): Shape-from-shading, viscosity solutions and edges. Numer. Math. 64, 323–353MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Rouy, E., Tourin, A. (1992): A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29, 867–884MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Oliensis, J., Dupuis, P. (1993): A global algorithm for shape-from-shading. In: Fourth international conference on computer vision. Proceedings. IEEE, Piscataway, NJ, pp. 692–701Google Scholar
  20. [20]
    Sagona, M. (2001): Numerical methods for degenerate Eikonal type equations and applications. Tesi di Dottorato. Universitá di Napoli “Federico II”, NaplesGoogle Scholar
  21. [21 ]
    Sagona, M., Seghini, A. (2001): An adaptive scheme on unstructured grids for the shape-from-shading problem. In [11], pp. 197–219Google Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • M. Falcone
    • 1
  • M. Sagona
    • 1
  • A. Seghini
    • 1
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly

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