An exact subdivision of the Radon transform and scanning with a positron ring camera.

It has been pointed out (Cormack 1978) that if a beam of finite width, defined in a certain way, is used in the two-dimensional reconstruction of an object, the beam defines a finite region of the Radon transform of the object. A novel method of scanning was suggested in which the regions of the transform defined by different positions of the beam fitted together almost exactly in such a way that most of the transform was sampled only once, and only a small fraction of the transform was either not sampled or was sampled twice. A natural question to ask is whether there are other methods of scanning which sample the transform in such a way that the regions of the transform defined by different positions of the beam fit together exactly. The purpose of this Note is to point out that this is accomplished in an ideal positron ring scanner, and that this is closely related to the treatment of such a scanner by Marr (1974).