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A viscosity method for Shape-from-Shading without boundary data

Emmanuel Prados 1 Fabio Camilli Olivier Faugeras
1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique de l'École normale supérieure, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : This report proposes a solution of the Lambertian Shape From Shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solutions. The power of our approach is twofolds: 1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Note that, in the previous SFS work of Rouy et al., Falcone et al. , Prados et al., the characterization of a viscosity solution and its computation require the knowledge of its values on the boundary of the image. This was quite unrealistic because in practice such values are not known. 2) it unifies the work of Rouy et al., Falcone et al., Prados et al., based on the notion of viscosity solutions and the work of Dupuis and Oliensis dealing with classical ($C^1$) solutions and value functions. Also, we generalize their work to the ``perspective SFS'' problem recently introduced by Prados and Faugeras. The notion of viscosity solutions described in this paper is obtained by slightly modifying the notion of singular viscosity solutions developped by Camilli and Siconolfi. We demonstrate the existence and the uniqueness of the solution for a class of Hamilton-Jacobi equations $$H(x,\nablau)=0,$$ in a bounded open domain. Some stability results are proved. Moreover, we show that this framework allows to characterize the classical discontinuous viscosity solutions by their ``minimums''. In this report, we also propose some algorithms which provide numerical approximations of these new solutions. These provably convergent algorithms are quite robust and do not necessarily require boundary data. Finally, we have successfully applied our SFS method to real images and we have suggested a number of real-life applications.
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Submitted on : Friday, May 19, 2006 - 9:17:03 PM
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  • HAL Id : inria-00070704, version 1



Emmanuel Prados, Fabio Camilli, Olivier Faugeras. A viscosity method for Shape-from-Shading without boundary data. RR-5296, INRIA. 2004, pp.35. ⟨inria-00070704⟩



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