Cone-beam reconstruction by the use of Radon transform intermediate functions

A mathematical framework is presented for cone-beam reconstruction by intermediate functions that are related to the three-dimensional Radon transform of the object being imaged. Cone-beam projection data are processed with a filter to form an intermediate function. The filter is a linear combination ah1(l) + bh2(l) of the ramp kernel h1(l) and the derivative functional h2(l). From the intermediate function, the reconstruction is completed with a convolution and backprojection, where the convolution filter is another linear combination ch1(l) + dh2(l) subject to the restriction ac + bd = 1. This formulation unifies and generalizes the important cone-beam formulas of Tuy [ SIAM J. Appl. Math.43, 546 ( 1983)], Smith [ IEEE Trans. Med. ImagingMI-4, 14 ( 1985)], and Grangeat [ Ph.D. dissertation ( Ecole Nationale Supérieure des Télécommunications, Paris, 1987)]. The appropriate values of a, b, c, d for these methods are derived.