A ring of positron annihilation detectors capable of 2 independent motions in the plane of the ring is considered. It can rotate around its center, and the whole ring may move so that its center traces a circular path. Provided certain parameters are chosen according to given formulas, data collected by such a detector ring can be reorganized (rebinned), in a computationally very inexpensive fashion, so that the data items in each bin are measurements along parallel lines spaced at small equal intervals. Efficient high resolution reconstruction algorithms that assume such data organization can therefore be applied to data collected by a moving positron ring detector. The method is demonstrated on a realistic example.