Abstract
Gardner [7] proved that with the exception of a simple class of nonparallel wedges, convex polygons in the plane are uniquely determined by one directed X-ray. This paper develops methods for reconstructing convex polygons in the plane from one directed X-ray. We show that nonsmooth points on the boundary of a convex body are located along rays where the derivative of the data have a jump discontinuity. Location of the nonsmooth points divides a convex polygon into a finite set of wedges. We prove uniqueness theorems and give algorithms for reconstructing nonparallel wedges from line integrals along four or more rays. Also, we characterize discrete data sets that are directed X-rays of both parallel and nonparallel wedges. Several examples of reconstructions are included.
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