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Unifying Approaches and Removing Unrealistic Assumptions in Shape from Shading: Mathematics Can Help

Emmanuel Prados 1 Olivier Faugeras 1
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 article 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 Shape from Shading work of Rouy et al. [rouy-tourin:92,lions-rouy-etal:93], Falcone et al. [falcone-sagona-etal:01], Prados et al. [prados-faugeras-etal:02,prados-faugeras:03], 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. [rouy-tourin:92,lions-rouy-etal:93], Falcone et al. [falcone-sagona-etal:01], Prados et al. [prados-faugeras-etal:02,prados-faugeras:03], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [dupuis-oliensis:94] dealing with classical solutions. Also, we generalize their work to the ``perspective Shape from Shading'' problem recently introduced by Prados and Faugeras [prados-faugeras:03]. Moreover this article introduces a ``generic'' formulation of the Shape from Shading problem. This ``generic'' formulation summarizes various (classical) formulations of the Lambertian Shape from Shading problem. In particular it unifies the orthographic and the perspective Shape from Shading problems. This ``generic'' formulation significantly simplifies the formalism of the problem. Thanks to this generic formulation, a single algorithm can be used to compute numerical solutions of all these previous Shape from Shading formulations. Finally we propose two algorithms which provide numerical approximations of the new weak solutions of the ``generic Shape from Shading'' problem. These provably convergent algorithms are quite robust and do not necessarily require boundary data.
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Emmanuel Prados, Olivier Faugeras. Unifying Approaches and Removing Unrealistic Assumptions in Shape from Shading: Mathematics Can Help. Proceedings of the 8th European Conference on Computer Vision, May 2004, Prague, Czech Republic, European Union. pp.141--154, ⟨10.1007/b97873⟩. ⟨inria-00394234⟩

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