Reduction of rank-reduced orientation-from-color problem with many unknown lights to two-image known-illuminant photometric stereo

How does shape from shading proceed when two or more differently colored light sources are present? For a uniformly colored Lambertian surface illuminated by a collection of point or extended light sources or inter reflections, with unknown directions and strengths, illumination varies spectrally with orientation from the surface. If light varies enough in color and direction, then surface orientation can be recovered by a kind of photometric stereo, but with a single image and many unknown lights. RGB values lie on an ellipsoid in color and the strengths and directions of three effective “lights” can be recovered from regression and by additional constraints derived from three estimates of light source tilt. However, it is quite likely that illumination color does not vary enough. In that case RGB points fill a planar 2D ellipse in color space. We give a result for the ellipse boundary that enables one to recover the strengths and angle between two effective lights producing the 2D shading. We then show that equations for the tilts of the linearly related three effective lights in 3D give an additional three independent constraints. Solving yields lights in 2D and hence in 3D as well. Thus the rank 2 orientation from color problem reduces to known light 2 image photometric stereo. Robust methods are used throughout