Unifying approaches and removing unrealistic assumptions in shape from shading: Mathematics can help

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 SFS work of Rouy et al. [23,15], Falcone et al. [8], Prados et al. [22,20], 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. [23,15], Falcone et al. [8], Prados et al. [22,20], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [6] dealing with classical (C1) solutions. Also, we generalize their work to the "perspective SFS" problem recently introduced by Prados and Faugeras [20]. Moreover this article introduces a "generic" formulation of the SFS problem. This "generic" formulation summarizes various (classical) formulations of the Lambertian SFS problem. In particular it unifies the orthographic and the perspective SFS 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 SFS formulations. Finally we propose two algorithms which provide numerical approximations of the new weak solutions of the "generic SFS" problem. These provably convergent algorithms are quite robust and do not necessarily require boundary data. © Springer-Verlag Berlin Heidelberg 2004.