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# Shape from Shading: a well-posed problem?

1 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique - ENS Paris, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS-PSL - 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 $1/r^2$ 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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Cited literature [1 references]

https://hal.inria.fr/inria-00070703
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Submitted on : Friday, May 19, 2006 - 9:16:46 PM
Last modification on : Thursday, March 17, 2022 - 10:08:32 AM
Long-term archiving on: : Sunday, April 4, 2010 - 8:13:13 PM

### Identifiers

• HAL Id : inria-00070703, version 1

### Citation

Emmanuel Prados, Olivier Faugeras, Fabio Camilli. Shape from Shading: a well-posed problem?. RR-5297, INRIA. 2004, pp.55. ⟨inria-00070703⟩

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