The eikonal equation: some results applicable to computer vision

In this paper we investigate certain first order partial differential equations which formulate the relationship between the light reflected from a surface and its shape. Particular emphasis is given to eikonal equations. Two results are presented. First, we prove that a special type of eikonal equation has only one convex and positive C2 solution in some neighborhood of a singular point. Using this result, we show that a restricted form of this equation has exactly two solutions. These results have application in scanning electron microscopy.