X-Ray Problems for Point Sources

Abstract

Let K be a plane convex set, assumed compact with non-empty interior. Let P{ and P2 be distinct known points in the plane, and let j](9) be the length of the chord of K through P, that makes angle 6 with some fixed axis. We call f, the chord function of K at P,. Under what circumstances is it possible to determine the set K from the functions / , and /2? If the points P, are exterior to K, this is the X-ray problem of Hammer [5] for point sources: how many pictures taken with point sources of X-rays are required to permit the reconstruction of a convex body of uniform density? (The case of Hammer's X-ray problem for parallel beams of X-rays has been considered by Gardner and McMullen [4].) We shall show here that, except in certain awkward cases, if the line / through P{ and P2 is known to intersect the interior of K, then K is uniquely determined by fy(Q) and /2(0). This generalizes a result of Rogers [7] that if f{(0) = /2(0) for all 0 then K must be centrally symmetric about the mid-point of Px P2 (see also Larman and Tamvakis [6] for variations of this). For the 'parallel beam' problem Gardner and McMullen [4] showed that two convex sets could be distinguished by X-ray photographs taken in certain sets of directions, but provided no method of reconstructing a set from its X-ray photographs. Using our method for the 'point source' problem, however, it would be possible, at least theoretically, to reconstruct a set K to any required degree of accuracy from its chord functions at Px and P2; in this sense our method is constructive. Production of a practical reconstruction algorithm and the error analysis, whilst feasible, would be complicated since this would involve the solution of non-linear simultaneous equations and equations defined by the limit of an iterative procedure. It is appropriate to mention the related work that is being done in the field of X-ray tomography, where the problem is to reconstruct an object of varying density as accurately as possible from a large number of X-ray photographs. The paper of Finch, Smith, and Solmon [3] describes some of the most recent theoretical work using point sources of X-rays (the 'divergent beam' problem). However, the problems considered here are of a different nature and are probably of less practical application. The main proofs used here are related to the method of Rogers [7] and also employ a lemma on the behaviour of certain real sequences. We remark at the outset that given a point P, and the chord function / of some convex set K, it is clear from / whether or not P is an interior point of K: the point P is interior to K if and only if/ is continuous and non-vanishing. It is more awkward to distinguish between an exterior point and a boundary point of K. If f(0) vanishes for 0 in some proper interval then either P is exterior to K or P is a boundary point of K at which K has more than one line of support. If P is a boundary point of K then the reconstruction of K from / , modulo reflection in P, is trivial, and the chord function