"Perspective shape from shading" and viscosity solutions

Abstract

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.

Shape from Shading for Rough Surfaces: Analysis of the Oren-Nayar Model

“…for realistic imagery. However, recently, more realistic concepts such as perspective cameras [14,15,21] and nonLambertian reflectance models [3,19] found their way into research and led to considerable progress in the field. In particular, considering more realistic camera models and more realistic reflectance models combined with a physically motivated light attenuation term [15] turned out to be useful from both a theoretical and practical viewpoint [2,25].…”

Shape from Shading: Recognizing the Mountains through a Global View

“…However, their method does not work for saddle (shoulder) points, which is a serious limitation. Prados and Faugeras [12] assumed that boundary conditions as well as heights at singular points were given, but these data are generally inaccessible. Their recent work [13] considered the effect of the distance between the light source and the surface and a monotonous scheme was obtained.…”

Perspective shape-from-shading by fast marching

“…When comparing the perspective Fast Marching method with the perspective algorithm of [14], perspective Fast Marching has a prominent advantage of runtime. In [14], the famous synthetic Vase image without noise required about 1000 iterations to recover and about 2000 iterations when synthetic noise was added; a synthetic Mozart face image required about 4000 iterations (without noise).…”

“…Recently, [20] and [14] suggested the use of a perspective projection model. Both papers introduced the perspective image irradiance equation simultaneously.…”

“…Both papers introduced the perspective image irradiance equation simultaneously. In [20], the suggested computational algorithm is a very preliminary one, while both algorithms ([20], [14]) are slow and were demonstrated on synthetic images only. …”