A viscosity solution method for shape-from-shading without image boundary data
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 2, pp. 393-412.

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 Vision 2351 (2002) 790-804; Prados and Faugeras, IEEE Comput. Soc. Press 2 (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.

DOI : https://doi.org/10.1051/m2an:2006018
Classification : 35D99,  62H35,  65N06,  65N12,  68T45
Mots clés : shape-from-shading, boundary data, unification of SFS theories, singular viscosity solutions, states constraints
@article{M2AN_2006__40_2_393_0,
author = {Prados, Emmanuel and Camilli, Fabio and Faugeras, Olivier},
title = {A viscosity solution method for shape-from-shading without image boundary data},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {393--412},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
year = {2006},
doi = {10.1051/m2an:2006018},
zbl = {1112.49025},
mrnumber = {2241829},
language = {en},
url = {http://www.numdam.org/item/M2AN_2006__40_2_393_0/}
}
Prados, Emmanuel; Camilli, Fabio; Faugeras, Olivier. A viscosity solution method for shape-from-shading without image boundary data. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 2, pp. 393-412. doi : 10.1051/m2an:2006018. http://www.numdam.org/item/M2AN_2006__40_2_393_0/

[1] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhauser, Boston (1997). | MR 1484411 | Zbl 0890.49011

[2] G. Barles, An approach of deterministic control problems with unbounded data. Ann. I. H. Poincaré 7 (1990) 235-258. | Numdam | Zbl 0717.49021

[3] G. Barles, Solutions de Viscosité des Equations de Hamilton-Jacobi. Springer-Verlag, Paris (1994). | Zbl 0819.35002

[4] 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. | Zbl 0691.49028

[5] I. Barnes and K. Zhang, Instability of the eikonal equation and shape-from-shading. ESAIM: M2AN 34 (2000) 127-138. | Numdam | Zbl 0973.35017

[6] F. Camilli and A. Siconolfi, Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems. Indiana U. Math. J. 48 (1999) 1111-1132. | Zbl 0939.49019

[7] F. Camilli and A. Siconolfi, Nonconvex degenerate Hamilton-Jacobi equations. Math. Z. 242 (2002) 1-21. | Zbl 1058.35063

[8] I. Capuzzo-Dolcetta and P.-L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643-68. | Zbl 0702.49019

[9] F.H. Clarke, Optimization and Nonsmooth Analysis. SIAM, Classics in Applied Mathematics 5, Philadelphia (1990). | MR 1058436 | Zbl 0696.49002

[10] M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. | Zbl 0599.35024

[11] 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. | Zbl 0807.49027

[12] 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.

[13] B.K. Horn and M.J. Brooks, Eds., Shape From Shading. The MIT Press (1989). | MR 1062877

[14] 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. | EuDML 84047 | Numdam | Zbl 0701.35052

[15] 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. | Zbl 0842.35019

[16] R. Kimmel, K. Siddiqi, B.B. Kimia and A. Bruckstein, Shape from shading: Level set propagation and viscosity solutions. Int. J. Comput. Vision 16 (1995) 107-133.

[17] P.-L. Lions, Generalized Solutions of Hamilton-Jacobi Equations. Res. Notes Math. 69. Pitman Advanced Publishing Program, London (1982). | Zbl 0497.35001

[18] P.-L. Lions, E. Rouy and A. Tourin, Shape-from-shading, viscosity solutions and edges. Numer. Math. 64 (1993) 323-353. | EuDML 133708 | Zbl 0804.68160

[19] 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. | Zbl 1059.35028

[20] 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.

[21] 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. Press 2 (2003) 826-831. | Zbl 1039.68702

[22] E. Prados and O. Faugeras, A generic and provably convergent shape-from-shading method for orthographic and pinhole cameras. Int. J. Comput. Vision 65 (2005) 97-125.

[23] 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. | Zbl 1039.68702

[24] 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). | MR 2283609

[25] E. Rouy and A. Tourin, A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29 (1992) 867-884. | Zbl 0754.65069

[26] H.M. Soner, Optimal control with state space constraints. SIAM J. Control Optim 24 (1986): Part I: 552-562, Part II: 1110-1122. | Zbl 0619.49013

[27] 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.

[28] 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. Press 2 (2003) 862-869.

[29] D. Tschumperlé, PDE's Based Regularization of Multivalued Images and Applications. Ph.D. Thesis, University of Nice-Sophia Antipolis (2002).

[30] R. Zhang, P.-S. Tsai, J.-E. Cryer and M. Shah, Shape from shading: A survey. IEEE T. Pattern Anal. 21 (1999) 690-706. | Zbl 1316.94019