A note on existence and uniqueness in shape from shading

Shape recovery of a smooth Lambertian surface illuminated by one, two, or three distant light sources is discussed. In the case of three light-sources, it is shown that the corresponding system of three first-order nonlinear partial differential equations has a unique solution. In the case of one overhead light-source, the existence and uniqueness of solutions to the corresponding eikonal equation are discussed. With regard to existence, two classes of shading patterns are shown for which no genuine shapes exist. In connection with uniqueness, circularly symmetric eikonal equations are revealed for which there exist circularly symmetric and non-circularly symmetric smooth solutions of special form