# Application of the theory of the viscosity solutions to the Shape From Shading problem

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 considered as an ill-posed problem which is therefore difficult to solve. In order to understand the difficulty of this problem and to design reliable and relevant solutions, we propose a rigorous approach based on the notion of viscosity solutions. In particular, we systematically prove the existence and uniqueness of the solution; if necessary (when uniqueness does not hold) we characterize the solutions. We also demonstrate the convergence of our algorithms. After having considered and fully exploited the (partial differential) equations provided by the classical modeling of the Shape From Shading problem, we propose and study some new equations arising from modelings which are more realistic than the ones considered in the Shape From Shading literature. In particular, this allows us to demonstrate that, with more realistic modelings, the Shape From Shading problem is generally well-posed. In effect, we prove that the classical version of the Shape From Shading problem is ill-posed because of an over-simplification in the modeling. In this work, we also propose an extension of the notion of the singular viscosity solutions developed recently by Camilli and Siconolfi. This extension allows to propose a new characterization of the discontinuous viscosity solutions. This new mathematical framework also allows to unify the various theoretical results that have been obtained in the Shape From Shading area.
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https://tel.archives-ouvertes.fr/tel-00007916
Submitted on : Tuesday, January 4, 2005 - 11:14:55 PM
Last modification on : Thursday, October 29, 2020 - 3:01:35 PM
Long-term archiving on: : Friday, April 2, 2010 - 8:50:16 PM

### Identifiers

• HAL Id : tel-00007916, version 1

### Citation

Emmanuel Prados. Application of the theory of the viscosity solutions to the Shape From Shading problem. Human-Computer Interaction [cs.HC]. Université Nice Sophia Antipolis, 2004. English. ⟨tel-00007916⟩

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