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Shape from Shading: a well-posed problem?

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 : Shape From Shading is known to be an ill-posed problem. Contrary to the previous work, we show here that if we model the problem in a more realistic way than it is usually done (we take into account the 1/r2 attenuation term of the lighting), Shape From Shading can be 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 regularity assumptions and without the knowledge of the heights of the solution at the local "minima". More precisely, in this report we formulate the problem as that of solving a new PDE, we develop a complete mathematical study of this equation (existence and uniqueness of the solution) and we design a new provably convergent numerical method. Finally, we test our new SFS method on various synthetic images and on our database of real images of faces, with success.
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Emmanuel Prados, Olivier Faugeras. Shape from Shading: a well-posed problem?. IEEE Conference on Computer Vision and Pattern Recognition, CVPR'05, Jun 2005, San Diego, United States. pp.870-877, ⟨10.1109/CVPR.2005.319⟩. ⟨inria-00394230⟩

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