Existence and Uniqueness in Shape from Shading

For the image of a smooth surface object fully contained within the field of view and illuminated in an arbitrary direction, this paper discusses the ekistence and uniqueness of the conditions for solving a shape-from-shading problem under the conditions that the Fourier series expansion of the image intensity contains only zero and first order terms in a polar coordinate system. Three theorems are established, one for the ekistence and two for the uniqueness of z-axis symmetric shape from shading.