A mathematical and algorithmic study of the Lambertian SFS problem for orthographic and pinhole cameras

This report proposes a mathematical and algorithmic study of the Lambertian SFS problem for orthographic and pinhole cameras. Our approach is based upon the notion of viscosity solutions of Hamilton-Jacobi equations. This approach provides a mathematical framework in which we can prove the well-posedness of the problem (proof of the existence of a solution and characterization of all solutions). This mathematical approach allows also to prove the correctness of our methods. In particular, we describe a simple monotonous stability condition for the studied decentered schemes and we prove the convergence of their solutions toward the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Also, we show that this theory naturally applies to the SFS problems. Our work extends previous work in the SFS area in three directions. First, it models the camera both as orthographic and as perspective (pinhole), i.e whereas most authors assume an orthographic projection (see [20] for a panorama of the SFS problem up to 1989 and [43, 28] for more recent surveys); thus we extend the applicability of shape from shading methods to more realistic acquisition models. In particular it extends the work of [37, 38] and [39]. Also, by introducing a generic Hamiltonian, we work in a general framework allowing to deal with both models, thereby simplifying the formalization of the problem. Second, it gives some novel mathematical formulations of this problem yielding new partial di erential equations. Results about the existence and uniqueness of their solution are also obtained. Third, it allows us to come up with two new generic algorithms for computing numerical approximations of the "continuous" solution (of the generic SFS problem ) as well as a proof of their convergence toward that solution. Moreover, our two generic algorithms are able to deal with discontinuous images as well as images containing black shadows. Also, http://www-sop.inria.fr/odyssee/team/Emmanuel.Prados/ y http://www-sop.inria.fr/odyssee/team/Olivier.Faugeras/ 2 E. Prados & O. Faugeras one of the algorithms we propose in this report, seems to be the most e ective iterative algorithm of the SFS literature. From a more general viewpoint, our numerical results follow from a new method for solving Hamilton-Jacobi-Bellman equations. We propose two decentered nite di erence schemes. We detail the proofs of the stability and the consistency of these schemes, and the proof of the convergence of their associated algorithms. Key-words: Shape from Shading, Lambertian re ectance, pinhole camera, orthographic and perspective projection, uni cation of various modelizations, black shadows, discontiuous images, noise robustness, fast SFS algorithm, viscosity solutions, existence and uniqueness of a solution, characterization of the solutions, Hamilton-Jacobi-Bellman equations, monotonous approximation schemes, decentered schemes for HJB equations, nite di erence schemes, stability and convergence of schemes, iterative numerical algorithms.