Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T03:14:19.513Z Has data issue: false hasContentIssue false
  • English
  • Français

A viscosity solution method for Shape-From-Shadingwithout image boundary data

Published online by Cambridge University Press:  21 June 2006

Emmanuel Prados ,
Fabio Camilli  and
Olivier Faugeras
Emmanuel Prados
Affiliation:
Perception Team, INRIA Rhône-Alpes, France. Emmanuel.Prados@inrialpes.fr
Fabio Camilli
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università dell'Aquila, Italy. camilli@ing.univaq.it
Olivier Faugeras
Affiliation:
Odyssée Lab., INRIA Sophia Antipolis, France. Olivier.Faugeras@sophia.inria.fr
Get access

Abstract

In this paper we propose a solution of the Lambertian shape-from-shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solution. The power of our approach is twofolds: (1) it defines a notion of weak solutions (in the viscosity sense) which does notnecessarily require boundary data. Moreover, it allows to characterize theviscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal.29 (1992) 867–884], [Lions et al., Numer. Math.64 (1993) 323–353], [Falcone and Sagona, Lect. Notes Math.1310 (1997) 596–603],[Prados et al., Proc. 7th Eur. Conf. Computer Vision2351 (2002) 790–804; Prados and Faugeras, IEEE Comput. Soc. Press2 (2003) 826–831], based on the notion of viscosity solutions and the work of [Dupuis and Oliensis, Ann. Appl. Probab.4 (1994) 287–346] dealing with classical solutions.

Keywords

Shape-from-shadingboundary dataunification of SFS theoriessingular viscosity solutionsstatesconstraints.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 2 , March 2006 , pp. 393 - 412
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhauser, Boston (1997).
Barles, G., An approach of deterministic control problems with unbounded data. Ann. I. H. Poincaré 7 (1990) 235–258. CrossRef
G. Barles, Solutions de Viscosité des Equations de Hamilton–Jacobi. Springer–Verlag, Paris (1994).
Barles, G. and Perthame, B., Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Opt. 21 (1990) 21–44. CrossRef
Barnes, I. and Zhang, K., Instability of the eikonal equation and shape-from-shading. ESAIM: M2AN 34 (2000) 127–138. CrossRef
Camilli, F. and Siconolfi, A., Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems. Indiana U. Math. J. 48 (1999) 1111–1132. CrossRef
Camilli, F. and Siconolfi, A., Nonconvex degenerate Hamilton-Jacobi equations. Math. Z. 242 (2002) 1–21. CrossRef
Capuzzo-Dolcetta, I. and Lions, P.-L., Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643–68. CrossRef
F.H. Clarke, Optimization and Nonsmooth Analysis. SIAM, Classics in Applied Mathematics 5, Philadelphia (1990).
Crandall, M.G. and Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1–42. CrossRef
Dupuis, P. and Oliensis, J., An optimal control formulation and related numerical methods for a problem in shape reconstruction. Ann. Appl. Probab. 4 (1994) 287–346. CrossRef
Falcone, M. and Sagona, M., An algorithm for the global solution of the Shape-From-Shading model, in Proceedings of the International Conference on Image Analysis and Processing. Lect. Notes Math. 1310 (1997) 596–603.
B.K. Horn and M.J. Brooks, Eds., Shape From Shading. The MIT Press (1989).
Ishii, H., A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1989) 105–135.
Ishii, H. and Ramaswamy, M., Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients. Commun. Part. Diff. Eq. 20 (1995) 2187–2213. CrossRef
Kimmel, R., Siddiqi, K., Kimia, B.B. and Bruckstein, A., Shape from shading: Level set propagation and viscosity solutions. Int. J. Comput. Vision 16 (1995) 107–133. CrossRef
P.-L. Lions, Generalized Solutions of Hamilton–Jacobi Equations. Res. Notes Math. 69. Pitman Advanced Publishing Program, London (1982).
Lions, P.-L., Rouy, E. and Tourin, A., Shape-from-shading, viscosity solutions and edges. Numer. Math. 64 (1993) 323–353. CrossRef
Malisoff, M., Bounded-from-below solutions of the Hamilton-Jacobi equation for optimal control problems with exit times: vanishing Lagrangians, eikonal equations, and shape-from-shading. NoDEA: Nonlinear Differ. Equ. Appl. 11 (2004) 95–122. CrossRef
J. Oliensis and P. Dupuis, Direct method for reconstructing shape from shading, in Proceedings of SPIE Conf. 1570 on Geometric Methods in Computer Vision (1991) 116–128.
Prados, E. and Faugeras, O., Perspective shape-from-shading, and viscosity solutions, in Proceedings of the 9th International Conference on Computer Vision (Nice 2003). IEEE Comput. Soc. Press 2 (2003) 826–831.
Prados, E. and Faugeras, O., A generic and provably convergent shape-from-shading method for orthographic and pinhole cameras. Int. J. Comput. Vision 65 (2005) 97–125. CrossRef
E. Prados, O. Faugeras and E. Rouy, Shape from shading and viscosity solutions, in Proceedings of the 7th European Conference on Computer Vision (Copenhagen 2002), Springer-Verlag 2351 (2002) 790–804.
E. Prados, F. Camilli and O. Faugeras, A unifying and rigorous shape from shading method adapted to realistic data and applications. J. Math. Imaging Vis. (2006) (to appear).
Rouy, E. and Tourin, A., A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29 (1992) 867–884. CrossRef
H.M. Soner, Optimal control with state space constraints. SIAM J. Control Optim 24 (1986): Part I: 552–562, Part II: 1110–1122.
H.J. Sussmann, Uniqueness results for the value function via direct trajectory-construction methods, in Proceedings of the 42nd IEEE Conference on Decision and Control 4 (2003) 3293–3298.
Tankus, A., Sochen, N. and Yeshurun, Y., A new perspective [on] Shape-From-Shading, in Proceedings of the 9th International Conference on Computer Vision (Nice 2003). IEEE Comput. Soc. Press 2 (2003) 862–869.
D. Tschumperlé, PDE's Based Regularization of Multivalued Images and Applications. Ph.D. Thesis, University of Nice-Sophia Antipolis (2002).
Zhang, R., Tsai, P.-S., Cryer, J.-E. and Shah, M., Shape from shading: A survey. IEEE T. Pattern Anal. 21 (1999) 690–706. CrossRef