Uniqueness in shape from shading revisited

We analyse the problem of representing solutions of first-order partial differential equations in terms of complete integrals and envelopes. In this context, we revisit the uniqueness results already existing in the shape-from-shading literature that concern eikonal equations corresponding to the images of a Lambertian hemi-sphere and a Lambertian plane. We show that the approach adopted by Brooks in [2, 3] is incomplete and subsequently re-establish its uniqueness claims.