The shading isophotes: Model and methods for Lambertian planes and a point light

Structure-from-Motion (SfM) and Shape-from-Shading (SfS) are complementary classical approaches to 3D vision. Broadly speaking, SfM exploits geometric primitives from textured surfaces and SfS exploits pixel intensity from the shading image. We propose an approach that exploits virtual geometric primitives extracted from the shading image, namely the level-sets, which we name shading isophotes. Our approach thus combines the strength of geometric reasoning with the rich shading information. We focus on the case of untextured Lambertian planes of unknown albedo lit by an unknown Point Light Source (PLS) of unknown intensity. We derive a comprehensive geometric model showing that the unknown scene parameters are in general all recoverable from a single image of at least two planes. We propose computational methods to detect the isophotes, to reconstruct the scene parameters in closed-form and to refine the results densely using pixel intensity. Our methods thus estimate light source, plane pose and camera pose parameters for untextured planes, which cannot be achieved by the existing approaches. We evaluate our model and methods on synthetic and real images.