Recent uniqueness results in shape from shading

Abstract

The main purpose of this paper is to discuss briefly two topics. The first one is to show that Sneddon’s claim ([9, Section 7 pp. 61]) about representability of any solution to a given first-order partial differential equation in terms of either a complete or a general or a singular integral is erroneous. The literature on complete integrals is a bewildering collection of incomplete and false statements (see e.g. Dou [6] or [9]). Recent results by Chojnacki [3] and Kozera [8] shed new light on this topic and fill a gap in the literature. The second goal of this paper is to critically inspect uniqueness results (see Brooks [1, 2]) concerning the images of a Lambertian hemisphere and a Lambertian plane, which resort to Sneddon’s erroneous assertion and as such are invalid. Finally, we adopt a different approach so that the results claimed in [1, 2], subject to minor reformulations, become valid. For a more detailed analysis an interested reader is also referred to [3] or [8].