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A viscosity solution method for shape-from-shading without image boundary data. (English) Zbl 1112.49025

Summary: 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 [E. Rouy and A. Tourin, SIAM J. Numer. Anal. 29, 867–884 (1992; Zbl 0754.65069)], [P.-L. Lions et al., Numer. Math. 64, 323–353 (1993; Zbl 0804.68160)], [M. Falcone and M. Sagona [Lect. Notes Math. 1310, 596–603 (1997)], [E. Prados et al, Proc. 7th Eur. Conf. Computer Vision 2351, 790–804, (2002; Zbl 1039.68702); E. Prados and O. Faugeras [IEEE Comput. Soc. Press 2, 826–831 (2003)], based on the notion of viscosity solutions and the work of [P. Dupuis and J. Oliensis [Ann. Appl. Probab. 4, 287–346 (1994; Zbl 0807.49027)] dealing with classical solutions.

MSC:

49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
68T45 Machine vision and scene understanding
35D99 Generalized solutions to partial differential equations
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References:

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