Generalised Perspective Shape from Shading in Spherical Coordinates

In the last four decades there has been enormous progress in Shape from Shading (SfS) with respect to both modelling and numerics. In particular approaches based on advanced model assumptions such as perspective cameras and non-Lambertian surfaces have become very popular. However, regarding the positioning of the light source, almost all recent approaches still follow the simplest geometric configuration one can think of: The light source is assumed to be located exactly at the optical centre of the camera. In our paper, we refrain from this unrealistic and severe restriction. Instead we consider a much more general SfS scenario based on a perspective camera, where the light source can be positioned anywhere in the scene. To this end, we propose a novel SfS model that is based on a Hamilton-Jacobi equation (HJE) which in turn is formulated in terms of spherical coordinates. This particular choice of the modelling framework and the coordinate system comes along with two fundamental contributions: While on the modelling side, the spherical coordinate system allows us to derive a generalised brightness equation – a compact and elegant generalisation of the standard image irradiance equation to arbitrary configurations of the light source, on the numerical side, the formulation as Hamilton-Jacobi equation enables us to develop a specifically tailored variant of the fast marching (FM) method – one of the most efficient numerical solvers in the entire SfS literature. Results on synthetic and real-world data confirm our theoretical considerations. They clearly demonstrate the feasibility and efficiency of the generalised SfS approach.