At the present time, largely because of a breakthrough in radiology called com uted tomography, the attenuation of x-ray beams is measured in extremely sensitive quantitative ways, and the information from many x-ray sources is assembled and analyzed on a computer. In this situation mathematics can make significant contributions concerning the nature of the information conveyed by x-rays from many sources, the extent to which this information determines the object x-rayed, suitable configurations of sources, methods for using the data to build a detailed reconstruction of the object, etc. This article announces results on these topics for the divergent x-ray beam. The three-dimensionally divergent beam, or cone beam, presents new problems that do not appear in the two-dimensional, or fan beam, case. Until recently there was little need for mathematics in radiology. Films were examined individually, and by eye, and mathematics had little to offer to the procedure. The picture changed radically in the late 1960s with a breakthrough called computed tomography, in which the attenuation of the x-ray beam is measured in an extremely sensitive quantitative way, and the information from many sources is assembled and analyzed on a computer (1). In this new situation mathematics can make significant contributions concerning the nature of the total information conveyed by x-rays from many sources, the extent to which this information determines the object x-rayed, suitable configurations of sources, methods for using the data to build a detailed reconstruction of the object, etc.In the initial device for computed tomography, the celebrated EMI scanner (1), a parallel x-ray beam was used, and two-dimensional cross sections of the object were reconstructed. The mathematical theory of the parallel beam x-ray transform is developed in refs. 2 and 3. In the current second generation of scanners two-dimensional cross sections are still reconstructed, but a two-dimensionally divergent x-ray beam is used in place of the parallel beam in order to allow faster scan times.At the present time there has arisen the need for dealing with a three-dimensionally divergent beam because of the extremely fast scan times required in the reconstruction of moving objects. A problem of major interest, for example, is the three-dimensional reconstruction of a beating heart. With human patients the x-ray data for such reconstructions must be collected during the fraction of a second in which heart movement is insignificant. This is impracticable with a succession of two-dimensionally divergent beams.In the two-dimensional case, practical reconstruction formulas for the divergent beam transform have been obtained by falling back upon known formulas for the parallel beam transform (4, 5), but even in the two-dimensional case very little has been known of the required mathematical theory. In the three-dimensional case, even such formulas have not been known. Moreover, the three-dimensional case differs radically from the former. From a given s...
