Shape from shading: a well-posed problem?

Shape from shading is known to be an ill-posed problem. We show in this paper that if we model the problem in a different way than it is usually done, more precisely by taking into account the 1/r² attenuation term of the illumination, shape from shading becomes completely well-posed. Thus the shading allows to recover (almost) any surface from only one image (of this surface) without any additional data (in particular, without the knowledge of the heights of the solution at the local intensity "minima", contrary to [P. Dupuis et al. (1994), E. Prados et al. (2004), B. Horn (1986), E. Rouy et al. (1992), R. Kimmel et al. (2001)]) and without regularity assumptions (contrary to [J. Oliensis et al. (1993), R. Kimmel et al. (1995)], for example). More precisely, we formulate the problem as that of solving a new partial differential equation (PDE), we develop a complete mathematical study of this equation and we design a new provably convergent numerical method. Finally, we present results of our new shape from shading method on various synthetic and real images.