# A viscosity solution method for Shape-From-Shading without image boundary data

Emmanuel Prados; Fabio Camilli; Olivier Faugeras

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

- Volume: 40, Issue: 2, page 393-412
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topPrados, Emmanuel, Camilli, Fabio, and Faugeras, Olivier. "A viscosity solution method for Shape-From-Shading without image boundary data." ESAIM: Mathematical Modelling and Numerical Analysis 40.2 (2006): 393-412. <http://eudml.org/doc/249696>.

@article{Prados2006,

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 not
necessarily require boundary data. Moreover, it allows to characterize the
viscosity 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.
},

author = {Prados, Emmanuel, Camilli, Fabio, Faugeras, Olivier},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Shape-from-shading; boundary data; unification of SFS theories; singular viscosity solutions; states
constraints.; states constraints},

language = {eng},

month = {6},

number = {2},

pages = {393-412},

publisher = {EDP Sciences},

title = {A viscosity solution method for Shape-From-Shading without image boundary data},

url = {http://eudml.org/doc/249696},

volume = {40},

year = {2006},

}

TY - JOUR

AU - Prados, Emmanuel

AU - Camilli, Fabio

AU - Faugeras, Olivier

TI - A viscosity solution method for Shape-From-Shading without image boundary data

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2006/6//

PB - EDP Sciences

VL - 40

IS - 2

SP - 393

EP - 412

AB -
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 not
necessarily require boundary data. Moreover, it allows to characterize the
viscosity 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.

LA - eng

KW - Shape-from-shading; boundary data; unification of SFS theories; singular viscosity solutions; states
constraints.; states constraints

UR - http://eudml.org/doc/249696

ER -

## References

top- M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhauser, Boston (1997). Zbl0890.49011
- G. Barles, An approach of deterministic control problems with unbounded data. Ann. I. H. Poincaré7 (1990) 235–258. Zbl0717.49021
- G. Barles, Solutions de Viscosité des Equations de Hamilton–Jacobi. Springer–Verlag, Paris (1994). Zbl0819.35002
- G. Barles and B. Perthame, Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Opt.21 (1990) 21–44. Zbl0691.49028
- I. Barnes and K. Zhang, Instability of the eikonal equation and shape-from-shading. ESAIM: M2AN34 (2000) 127–138. Zbl0973.35017
- F. Camilli and A. Siconolfi, Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems. Indiana U. Math. J.48 (1999) 1111–1132. Zbl0939.49019
- F. Camilli and A. Siconolfi, Nonconvex degenerate Hamilton-Jacobi equations. Math. Z.242 (2002) 1–21. Zbl1058.35063
- I. Capuzzo-Dolcetta and P.-L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc.318 (1990) 643–68. Zbl0702.49019
- F.H. Clarke, Optimization and Nonsmooth Analysis. SIAM, Classics in Applied Mathematics 5, Philadelphia (1990). Zbl0696.49002
- M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277 (1983) 1–42. Zbl0599.35024
- P. Dupuis and J. Oliensis, An optimal control formulation and related numerical methods for a problem in shape reconstruction. Ann. Appl. Probab.4 (1994) 287–346. Zbl0807.49027
- M. Falcone and M. Sagona, 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). Zbl0629.65125
- H. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci.16 (1989) 105–135. Zbl0701.35052
- H. Ishii and M. Ramaswamy, Uniqueness results for a class of Hamilton-Jacobi equations with singular coefficients. Commun. Part. Diff. Eq.20 (1995) 2187–2213. Zbl0842.35019
- R. Kimmel, K. Siddiqi, B.B. Kimia and A. Bruckstein, Shape from shading: Level set propagation and viscosity solutions. Int. J. Comput. Vision16 (1995) 107–133.
- P.-L. Lions, Generalized Solutions of Hamilton–Jacobi Equations. Res. Notes Math.69. Pitman Advanced Publishing Program, London (1982).
- P.-L. Lions, E. Rouy and A. Tourin, Shape-from-shading, viscosity solutions and edges. Numer. Math.64 (1993) 323–353. Zbl0804.68160
- M. Malisoff, 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. Zbl1059.35028
- 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.
- E. Prados and O. Faugeras, Perspective shape-from-shading, and viscosity solutions, in Proceedings of the 9th International Conference on Computer Vision (Nice 2003). IEEE Comput. Soc. Press2 (2003) 826–831.
- E. Prados and O. Faugeras, A generic and provably convergent shape-from-shading method for orthographic and pinhole cameras. Int. J. Comput. Vision65 (2005) 97–125.
- 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. Zbl1039.68702
- 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). Zbl1112.49025
- E. Rouy and A. Tourin, A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal.29 (1992) 867–884. Zbl0754.65069
- H.M. Soner, Optimal control with state space constraints. SIAM J. Control Optim24 (1986): Part I: 552–562, Part II: 1110–1122. Zbl0597.49023
- H.J. Sussmann, Uniqueness results for the value function via direct trajectory-construction methods, in Proceedings of the 42nd IEEE Conference on Decision and Control4 (2003) 3293–3298.
- A. Tankus, N. Sochen and Y. Yeshurun, A new perspective [on] Shape-From-Shading, in Proceedings of the 9th International Conference on Computer Vision (Nice 2003). IEEE Comput. Soc. Press2 (2003) 862–869.
- D. Tschumperlé, PDE's Based Regularization of Multivalued Images and Applications. Ph.D. Thesis, University of Nice-Sophia Antipolis (2002). Zbl1012.68786
- R. Zhang, P.-S. Tsai, J.-E. Cryer and M. Shah, Shape from shading: A survey. IEEE T. Pattern Anal.21 (1999) 690–706. Zbl1316.94019

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.