, Volume 25, Issue 3, pp 307-328
Date: 29 Aug 2006

A Unifying and Rigorous Shape from Shading Method Adapted to Realistic Data and Applications

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Abstract

We propose a new method for the Lambertian Shape From Shading (SFS) problem based on the notion of Crandall-Lions viscosity solution. This method has the advantage of requiring the knowledge of the solution (the surface to be reconstructed) only on some part of the boundary and/or of the singular set (the set of the points at maximal intensity). Moreover it unifies in an unique mathematical formulation the works of Rouy et al. [34, 50], Falcone et al. [21], Prados et al. [46, 48, 49], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [17] dealing with classical solutions and value functions. Also, it allows to generalize their results to the “perspective SFS” problem recently simultaneously introduced in [13,46,55].

While the theoretical part has been developed in [44], in this paper we give some stability results and we describe numerical schemes for the SFS based on this method. We construct provably convergent and robust algorithms. Finally, we apply our SFS method to real images and we suggest some real-life applications.

Emmanuel Prados holds a PhD in Computer Science and Image Processing at INRIA (National Research Institute in Computer Science and Control Theory) Sophia Antipolis (2004). He was a postdoctoral researcher at the Computer Science Department of the University of California at Los Angeles (2005). He is currently Research Scientist at INRIA Rhône-Alpes in the Perception laboratory.
His research interests includes some Computer Vision fields, in particular, the illumination and reflectance modeling and the scene reconstruction (Shape From Shading, multi-view reconstruction), as well as some Applied Mathematics fields: characterization of the solutions of Partial Differential Equations, notion of Viscosity Solutions of Hamilton-Jacobi equations, Optimal Control Theory, Differential Games, “Level Set” and “Fast Marching” methods, variational Framework.
He serves as a reviewer for major journals and conferences in image processing/computer vision and he was member of several computer vision conference and workshop Programme Commitees.
In January 2006, he received the accessit of the SPECIF Prize (Prize of the Best French PhD Thesis in Computer Science).
Fabio Camilli holds a PhD in Mathematics from the University of Rome “La Sapienza” (1996). He is currently Associate Professor at the University of l'Aquila (Italy). His research includes Partial Differential Equations theory, in particular Crandall-Lions viscosity solution theory for 1st and 2nd order Hamilton-Jacobi equations, Control theory, stability theory for deterministic and stochastic dynamical systems.
Olivier Faugeras is a graduate from the Ecole Polytechnique (1971).He holds a PhD in Computer Science and Electrical Engineering from the University of Utah (1976) and a Doctorate of Science from Paris VI University (1981).
He is currently Research Director at INRIA (National Research Institute in Computer Science and Control Theory), where he leads the Odyssée laboratory located in Sophia-Antipolis and Ecole Normale Supérieure, Paris. His research interests include the application of mathematics to computer and biological vision, shape representation and recognition, the use of functional imaging (MR, MEG, EEG) for understanding brain activity and in particular visual perception.
He has published extensively in archival Journals, International Conferences, has contributed chapters to many books and is the author of “Artificial 3-D Vision” published in 1993 by MIT Press and, with Quang-Tuan Luong and Theo Papadopoulo, of “The Geometry of Multiple Images” which appeared in March 2001, also at MIT Press.
He was an adjunct Professor from 1996 to 2001 in the Electrical Engineering and Computer Science Department of the Massachusetts Institute of Technology and a member of the AI Lab.
He is an Associate Editor of several international scientific Journals including Machine Vision and Applications, Videre, Image and Vision Computing. He has served as Associate Editor for IEEE PAMI from 1987 to 1990 and as co-Editor-in-Chief of the International Journal of Computer Vision from 1991 to 2004.
In April 1989 he received the “Institut de France—Fondation Fiat”award from the french Academy of Sciences for his work in Vision and Robotics.
In July 1998 he received the “France Telecom” award from the french Academy of Sciences for his work on Computer Vision and Geometry.
In November 1998 he was elected a member of the french Academy of Sciences.