Parallel X-ray tomography of convex domains as a search problem in two dimensions

In 1961, at A.M.S. Symposium on Convexity, P.C. Hammer proposed the following problem: how many X-ray pictures of a convex planar domain D must be taken to permit its exact reconstruction? Richard Gardner writes in his fundamental 2006 book 4] that X-rays in four different directions would do the job. The present paper points at the possibility that in certain asymptotical sense X-rays in only three different directions can be enough for approximate reconstruction of centrally symmetric convex domains. The accuracy of reconstruction would tend to become perfect in the limit, as the directions of the three X-rays change, all three converging to some given direction. The analysis leading to that conclusion is based on two lemmas of Section 1 and Pleijel type identity for parallel X-rays derived in Sections 2 and 3. These tools together supply a systemof two differential equations with respect to two unknown functions that describe the two branches of the domain boundary D. The system is easily resolved. The solution intended to provide a complete tomography reconstruction of D, happens however to depend on a two dimensional parameter, whose “real value” remains unknown. So tomography reconstruction of D becomes possible if a satisfactory approximation to that unknown “real value” can be found. In the last section a test procedure for the individual candidates for “approximate real value” of the parameter is described. A uniqueness theorem concerning tomography of circular discs is proved.