Circularly symmetric eikonal equations and non-uniqueness in computer vision

which arises naturally in wavefront analysis and in the development of special methods for integrating Hamilton’s equations (the Jacobi-Hamilton method), has long attracted the attention of physicists and mathematicians. More recently, there has been a resurgence of interest in the eikonal equation as a result of its applicability in an area of computer vision. One of the issues considered in the latter context has been that of determining whether or not a particular eikonal equation exhibits many solutions defined over a given domain. In this paper, we shall offer insight into this issue by presenting a non-uniqueness result of significance for the foundations of computer vision. A monochrome photograph of a smooth object will typically exhibit brightness variation, or shading. Of interest to researchers in computer vision is the problem of how object shape may be extracted from image shading. This shape-from-shading problem has been shown by Horn ([6]; 192 0022-241X/92 $3.00