SHADING WITHOUT SHAPE

A smooth object depicted in a monochrome image will often exhibit brightness variation, or shading. Of interest in computer vision is the inverse problem of how object shape may be recovered from such an image. This is referred to as the shape-from-shading problem. When the imaging conditions are such that an overhead point-source illuminates a smooth Lambertian surface, the problem may be formulated mathematically as that of finding a solution to an eikonal equation. In this paper, we seek images for which there are no corresponding object shapes. We are therefore concerned with the nonexistence of (bounded) solutions to certain eikonal equations. Specifically, we give a necessary and sufficient condition for a circularly-symmetric eikonal equation to admit exclusively unbounded solutions. In addition, we give a sufficient condition for an eikonal equation to have no solution. Examples are presented that elucidate the significance of these results for computer vision.