Journal of Mathematical Imaging and Vision

, Volume 25, Issue 3, pp 307–328 | Cite as

A Unifying and Rigorous Shape from Shading Method Adapted to Realistic Data and Applications



We propose a new method for the Lambertian Shape From Shading (SFS) problem based on the notion of Crandall-Lions viscosity solution. This method has the advantage of requiring the knowledge of the solution (the surface to be reconstructed) only on some part of the boundary and/or of the singular set (the set of the points at maximal intensity). Moreover it unifies in an unique mathematical formulation the works of Rouy et al. [34, 50], Falcone et al. [21], Prados et al. [46, 48, 49], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [17] dealing with classical solutions and value functions. Also, it allows to generalize their results to the “perspective SFS” problem recently simultaneously introduced in [13,46,55].

While the theoretical part has been developed in [44], in this paper we give some stability results and we describe numerical schemes for the SFS based on this method. We construct provably convergent and robust algorithms. Finally, we apply our SFS method to real images and we suggest some real-life applications.


Shape From Shading Hamilton-Jacobi equations viscosity solutions states constraints finite differences 


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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Emmanuel Prados
    • 1
  • Fabio Camilli
    • 2
  • Olivier Faugeras
    • 3
  1. 1.UCLA Vision Lab.USA
  2. 2.Dipartimento di Matematica Pura e ApplicataUniversità dell'AquilaItaly
  3. 3.Odyssée Lab. - INRIA Sophia AntipolisFrance

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