On complete integrals and uniqueness in shape from shading

In this paper, we analyze the problem of representing solutions of a first-order partial differential equation in terms of complete integrals. We show that some of the existing results referring to the representability problem are incomplete. Additionally, we analyze, in the context of complete integrals, the uniqueness problem for the shape recovery of a smooth Lambertian surface from an image obtained by illuminating this surface by an overhead, distant point light-source. Specifically, we revisit the uniqueness results already existing in the shape-from-shading literature that concern eikonal equations corresponding to the images of a Lambertian hemisphere and a Lambertian plane. We show that the latter results are incomplete and indicate how to fill the gaps in the corresponding proofs.