"Perspective Shape from Shading" and Viscosity Solutions
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چکیده
This article proposes a solution of the Lambertian shape from shading (SFS) problem in the case of a pinhole camera model (performing a perspective projection). Our approach is based upon the notion of viscosity solutions of HamiltonJacobi equations. This approach allows us to naturally deal with nonsmooth solutions and provides a mathematical framework for proving correctness of our algorithms. Our work extends previous work in the area in three aspects. First, it models the camera as a pinhole whereas most authors assume an orthographic projection (see [15] for a panorama of the SFS problem up to 1989 and [29, 17] for a recent survey), thereby extending the applicability of shape from shading methods to more realistic images. In particular it extends the work of [24] and [26]. Second, by adapting the brightness equation to the perspective problem, we obtain a new partial differential equation (PDE). Results about the existence and uniqueness of its solution are also obtained. Third, it allows us to come up with a new approximation scheme and a new algorithm for computing numerical approximations of the ”continuous” solution as well as a proof of their convergence toward that solution.
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تاریخ انتشار 2003