Isotropy, Reciprocity and the Generalized Bas-Relief Ambiguity

A set of images of a Lambertian surface under varying lighting directions defines its shape up to a three-parameter generalized bas-relief (GBR) ambiguity. In this paper, we examine this ambiguity in the context of surfaces having an additive non-Lambertian reflectance component, and we show that the GBR ambiguity is resolved by any non-Lambertian reflectance function that is isotropic and spatially invariant. The key observation is that each point on a curved surface under directional illumination is a member of a family of points that are in isotropic or reciprocal configurations. We show that the GBR can be resolved in closed form by identifying members of these families in two or more images. Based on this idea, we present an algorithm for recovering full Euclidean geometry from a set of uncalibrated photometric stereo images, and we evaluate it empirically on a number of examples.