Perspective Shape from Shading: Ambiguity Analysis and Numerical Approximations

Abstract

In this paper we study a modern, perspective model for shape from shading and its numerical approximation. We show that a new form of the classic concave/convex ambiguity is still present, although the model has been shown to be well-posed under particular assumptions. This analytical result is confirmed by various numerical tests. Moreover, we present convergence results for two iterative approximation schemes recently introduced in the literature. The first one is based on a finite difference discretization, whereas the second one is based on a semi-Lagrangian discretization. The convergence results are obtained in the general framework of viscosity solutions of the underlying partial differential equation. We also show that it is possible to obtain even in complex scenes results of reasonable quality. To this end we solve the constituting equation on a previously-segmented input image, where we use state constraints boundary conditions at the segment borders.

Generalised Perspective Shape from Shading with Oren-Nayar Reflectance

“…By assuming a Lambertian surface and the illumination given by a point light source located at the optical centre of the camera, they feature many well-posedness properties in contrast to their orthographic counterparts. In particular, the quadratic light fall-off due to inverse square law turned out to be very useful in this context [4,18]. Recently, such perspective SfS models with the light source located at the optical centre of the camera have been extended to non-Lambertian reflectance models [2,24] such as the aforementioned Oren-Nayar model [2].…”

Analysis of surface parametrizations for modern photometric stereo modeling

“…For example, Prados and Faugeras 2 derived a specific pinhole parametrization for the surface in the perspective scenario showing that their particular formulation made the SfS problem well-posed. Following, after few years Breuß et al 7 showed that it was actually not completely well-posed.…”

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

“…While for the Lambertian case Prados and Faugeras [15] were able to show that combining the assumption of a projective camera with a light attenuation term may turn the originally ill-posed SfS problem into a partially well-posed one (cf. [6]), no such thorough analysis has been performed for the advanced Oren-Nayar model so far. Moreover, to the best of our knowledge, there is only one work in the entire literature that presents such an in-depth study for a modern non-Lambertian SfS model: In [5], Breuß and Ju investigate important issues such as critical points and convexity in the context of SfS with the Phong model [25].…”