Impossible and ambiguous shading patterns

A smooth object depicted in a monochrome image will often exhibit brightness variation, or shading. A problem much studied in computer vision has been that of how object shape may be recovered from image shading. When the imaging conditions are such that an overhead point-source illuminates a smooth Lambertian surface, the problem may be formulated as that of finding a solution to an eikonal equation. This article will focus on the existence and uniqueness of such solutions, reporting recent results obtained. With regard to existence, shading patterns are exhibited for which there is no corresponding object shape. Specifically, a necessary and sufficient condition is presented for a circularly symmetric eikonal equation to admit exclusively unbounded solutions; additionally, a sufficient condition is given for an eikonal equation to have no solution whatsoever. In connection with uniqueness, we consider eikonal equations, defined over a disc, such that the Euclidean norm of the gradient of any solution is circularly symmetric, vanishes exactly at the disc center, and diverges to infinity as the circumference of the disc is approached. Contrary to earlier influential work, a class of such eikonal equations is shown to possess simultaneously circularly symmetric and noncircularly symmetric bounded smooth solutions.

Authors and Affiliations

1. Department of Computer Science, University of Adelaide, GPO Box 498, 5001, Adelaide, SA, Australia

   Michael J. Brooks

2. Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warszawa, Poland

   Wojciech Chojnacki

3. Discipline of Computer Science, School of Information Science and Technology, Flinders University of South Australia, GPO Box 2100, 5001, Adelaide, SA, Australia

   Wojciech Chojnacki

4. Department of Computer Science, University of Western Australia, 6009, Nedlands, WA, Australia

   Ryszard Kozera

Reprints and permissions