"Perspective Shape from Shading" and Viscosity Solutions

• Authors: E. Prados;O. Faugeras
• Venue: ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
• Year: 2003

This article proposes a solution of the Lambertian shapefrom shading (SFS) problem in the case of a pinhole cameramodel (performing a perspective projection). Our approachis based upon the notion of viscosity solutions of Hamilton-Jacobiequations. This approach allows us to naturallydeal with nonsmooth solutions and provides a mathematicalframework 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 whereasmost 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 ofshape from shading methods to more realistic images. Inparticular it extends the work of [24] and [26]. Second, byadapting the brightness equation to the perspective problem,we obtain a new partial differential equation (PDE).Results about the existence and uniqueness of its solutionare also obtained. Third, it allows us to come up with a newapproximation scheme and a new algorithm for computingnumerical approximations of the "continuous" solution aswell as a proof of their convergence toward that solution.