A Theory of Photometric Stereo for a Class of Diffuse Non-Lambertian Surfaces

Photometric stereo is a method of reconstructing a surface from the amount of light reflected by it. This is done by using prior knowledge of the surface reflectance to estimate the surface normal at all visible points. The theory of photometric stereo has been extensively developed for surfaces and illumination geometries that give rise to a Lambertian reflectance map. For non-Lambertian reflectance maps, the theory has been developed for specific cases, but a general theory has not been presented in the literature. In this paper, we propose a theory of photometric stereo for a large class of non-Lambertian reflectance maps. First, we review the different reflectance maps proposed in the literature for modeling reflection from real-world surfaces. From this, we obtain a mathematical class of reflectance maps to which the maps belong. Next, we show that three lights can be sufficient for a unique inversion of the photometric stereo equation for the entire class of reflectance maps. We also obtain a constraint on the positions of light sources for obtaining this solution. Next, we investigate the sufficiency of three light sources to estimate the surface normal and the illuminant strength. Finally, we address the issue of completeness of reconstruction. We show that if k lights are sufficient for a unique inversion, 2k lights are necessary for a complete inversion.