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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 Superieure des Telecommunications, Paris, 1987)]. The appropriate values of a, b, c, d for these methods are derived.

International audience; Cone beam x-ray tomography is based on the aquisition of two-dimensional projections for different positions of a cone beam x-ray source. Generally the x-ray source describes a single circle around the volume. We consider three different scanning modes: a sphere, a circle, and two orthogonal circles. We study the response of a cone beam reconstruction system defined as the weighted backprojection of the cone beam projections of an arbitrary impulse point. The spatial invariance of the system is discussed for the three acquisition geometries. Exact and approximate reconstruction methods based on three-dimensional convolution are then proposed. Some results obtained from both simulated and experimental data are presented and analyzed.

We compare two algorithms of three-dimensional (3D) cone-beam tomography: a 3D backprojection algorithm and a Radon algorithm, with a circular source trajectory. We recall the principle of these algorithms and show that when the source trajectory is circular, an exact reconstruction cannot be performed. We evaluate structures that cannot be detected and show that, for each algorithm, assumptions must be made about the object to estimate the missing information. Using simulated data, we measured the modulation transfer function of the two algorithms, evaluated the density resolution, and measured the artifacts due to missing information. We show that the 3D backprojection algorithm has good geometrical resolution along the axis of rotation and that, from the standpoint of density resolution, its useful limit is ±3°. We also show that it is sensitive to the lack of information in the horizontal planes. The useful aperture of the Radon algorithm is ±12°; for apertures lower than this limit, the algorithm is much less sensitive to the lack of information in the horizontal planes than the 3D backprojection algorithm. We present two examples of reconstructions that illustrate the limitations of these algorithms when applied to realistic samples and discuss the limits of using each algorithm when the source trajectory is circular.

A convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections. The formula is approximate but has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation. It reduces to the standard fan-beam formula in the plane that is perpendicular to the axis of rotation and contains the point source. The algorithm is applied to a mathematical phantom as an example of its performance.