Shape - from - shading on a cloudy day

In the classical shape-from-shading model, surface luminance depends primarily on the unit surface normal. However, under diffuse lighting conditions, such as the sky on a cloudy day, luminance depends primarily on the amount of sky that is visible from each surface element, with surface normal of secondary importance. This claim is formalized in terms of a dominating sky principle and a surface aperture function. An approximately functional constraint between surface luminance and aperture emerges. It is shown how to use this constraint to recover a depth map from an image efficiently. A curious difference from the classical shape-from-shading problem is uncovered. When one assumes a point light source, the local geometric constraints of the shape-from-shading problem lie along the surface. However, in the diffuse-lighting problem, the local geometric constraints are found in a visibility field, which is defined in the free space above the surface.