A VISCOSITY SOLUTION METHOD FOR SHAPE-FROM-SHADING WITHOUT IMAGE BOUNDARY DATA

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.

[1]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[2]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[3]  F. Camilli,et al.  Nonconvex degenerate Hamilton–Jacobi equations , 2002 .

[4]  Kewei Zhang,et al.  Instability of the eikonal equation and shape from shading , 2000 .

[5]  H. Ishii,et al.  Uniqueness results for a class of hamilton-jacobi equations with singular coefficients , 1995 .

[6]  Michael 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 , 2004 .

[7]  P. Dupuis,et al.  An Optimal Control Formulation and Related Numerical Methods for a Problem in Shape Reconstruction , 1994 .

[8]  Olivier D. Faugeras,et al.  Shape from Shading and Viscosity Solutions , 2002, ECCV.

[9]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[10]  Olivier D. Faugeras,et al.  A Unifying and Rigorous Shape from Shading Method Adapted to Realistic Data and Applications , 2006, Journal of Mathematical Imaging and Vision.

[11]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[12]  Maurizio Falcone,et al.  An Algorithm for the Global Solution of the Shape-from-Shading Model , 1997, ICIAP.

[13]  P. Lions,et al.  Shape-from-shading, viscosity solutions and edges , 1993 .

[14]  D. Tschumperlé.  PDE's based regularization of multivalued images and applications , 2002 .

[15]  Fabio Camilli,et al.  Maximal subsolutions for a class of degenerate Hamilton-Jacobi problems , 1999 .

[16]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[17]  Ping-Sing Tsai,et al.  Shape from Shading: A Survey , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  Paul Dupuis,et al.  Direct method for reconstructing shape from shading , 1991, Optics & Photonics.

[19]  Olivier D. Faugeras,et al.  A Generic and Provably Convergent Shape-from-Shading Method for Orthographic and Pinhole Cameras , 2005, International Journal of Computer Vision.

[20]  Hitoshi Ishii,et al.  A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations , 1989 .

[21]  Yehezkel Yeshurun,et al.  A new perspective [on] shape-from-shading , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[22]  G. Barles,et al.  An approach of deterministic control problems with unbounded data , 1990 .

[23]  G. Barles,et al.  Comparison principle for dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations , 1990 .

[24]  H. Soner.  Optimal control with state-space constraint I , 1986 .

[25]  E. Rouy,et al.  A viscosity solutions approach to shape-from-shading , 1992 .

[26]  H. Sussmann,et al.  Uniqueness results for the value function via direct trajectory-construction methods , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[27]  P. Lions,et al.  Hamilton-Jacobi equations with state constraints , 1990 .

[28]  Olivier D. Faugeras,et al.  Shape From Shading , 2006, Handbook of Mathematical Models in Computer Vision.

[29]  Alfred M. Bruckstein,et al.  Shape from shading: Level set propagation and viscosity solutions , 1995, International Journal of Computer Vision.

[30]  Olivier D. Faugeras,et al.  "Perspective shape from shading" and viscosity solutions , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.