On Uniqueness of Solutions of the Three-Light-Source Photometric Stereo: Conditions on Illumination Configuration and Surface Reflectance

This paper is concerned with photometric methods using three images with different lighting direction to obtain shape information of an object. Such methods are based on the photometric equation that relates the normal of the object surface to the triplet of the image brightness. This paper discusses the issue of whether the surface normal and the orientation of the 3-vector formed by the image brightness triplet is one-to-one in the equation. Several types of photometric methods require this relation to be one-to-one. We mainly consider the case where the reflectance map is an increasing function of the angle between the surface normal and the illuminant direction. We first point out that even in this simple case, it is possible that the relation is not one-to-one. Then we derive several sufficient conditions on the reflectance as well as the illumination configuration for the one-to-one relation.