Belief Propagation with Directional Statistics for Solving the Shape-from-Shading Problem

The Shape-from-Shading [SfS] problem infers shape from reflected light, collected using a camera at a single point in space only. Reflected light alone does not provide sufficient constraint and extra information is required; typically a smoothness assumption is made. A surface with Lambertian reflectance lit by a single infinitely distant light source is also typical. We solve this typical SfS problem using belief propagation to marginalise a probabilistic model. The key novel step is in using a directional probability distribution, the Fisher-Bingham distribution. This produces a fast and relatively simple algorithm that does an effective job of both extracting details and being robust to noise. Quantitative comparisons with past algorithms are provided using both synthetic and real data.