Feasible cone beam scanning methods for exact reconstruction in three-dimensional tomography

Abstract

Feasible cone beam scanning methods that permit exact reconstruction in three-dimensional tomography are proposed, and their performances are investigated using the theory developed by Kirillov [Sov. Math. Dokl. 2, 268 (1961)], Tuy [SIAM J. Appl. Math. 43, 546 (1983)], and Smith [IEEE Trans. Med. Imaging MI-4, 14 (1985)]. The proposed methods move an apex of a cone on two circles that are perpendicular to each other (orthogonal scan) or on a few twists of a helical curve (helical scan). The obtainable information, the feasibility of reconstruction, and the redundancy of measurement by some typical cone beam scanning methods are analyzed using the relationship between the cone beam projections and the three-dimensional Radon transform. Consequently it is found that the proposed methods can obtain the complete information about the three-dimensional Fourier transform of an object with less redundant measurement. Moreover, based on the analysis, a practically implementable reconstruction algorithm suitable for the proposed methods is derived. Experiments performed with both numerical and real x-ray projections demonstrate the validity of the proposed scanning methods and the reconstruction algorithm. Especially when a large cone angle is used, the proposed methods greatly improve the spatial resolution and the quantitative accuracy of reconstruction as compared with the method of circular apex motion.