Shape from shading as a partially well-constrained problem

For general objects, and for illumination from a general direction, we study the constraints on shape imposed by shading. Assuming generalized Lambertian reflectance, we argue that, for a typical image, shading determines shape essentially up to a finite ambiguity. Thus regularization is often unnecessary, and should be avoided. More conjectural arguments imply that shape is typically determined with little ambiguity. However, it is pointed out that the degree to which shape is constrained depends on the image. Some images uniquely determine the imaged surface, while, for others, shape can be uniquely determined over most of the image, but infinitely ambiguous in small regions bordering the image boundary, even though the image contains singular points. For these images, shape from shading is a partially well-constrained problem. The ambiguous regions may cause shape reconstruction to be unstable at the image boundary. Our main result is that, contrary to previous belief, the image of the occluding boundary does not strongly constrain the surface solution. Also, it is shown that characteristic strips are curves of steepest ascent on the imaged surface. Finally, a theorem characterizing the properties of generic images is presented.