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.
Keywords: 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 },
pages = {393--412},
year = {2006},
publisher = {EDP-Sciences},
volume = {40},
number = {2},
doi = {10.1051/m2an:2006018},
mrnumber = {2241829},
zbl = {1112.49025},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2006018/}
}
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 PY - 2006 SP - 393 EP - 412 VL - 40 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an:2006018/ DO - 10.1051/m2an:2006018 LA - en ID - M2AN_2006__40_2_393_0 ER -
%0 Journal Article %A Prados, Emmanuel %A Camilli, Fabio %A Faugeras, Olivier %T A viscosity solution method for shape-from-shading without image boundary data %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2006 %P 393-412 %V 40 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an:2006018/ %R 10.1051/m2an:2006018 %G en %F 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 , Tome 40 (2006) no. 2, pp. 393-412. doi: 10.1051/m2an:2006018
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