True three-dimensional cone-beam reconstruction (TTCR) algorithm

Derivation and implementation of a cone-beam reconstruction algorithm for nonplanar orbits

B.D. Smith (ibid., vol.MI-4, p.15-25, 1985; Opt. Eng., vol.29, p.524-34, 1990) and P. Grangeat (These de doctorat, 1987; Lecture Notes in Mathematics 1497, p.66-97, 1991) derived a cone-beam inversion formula that can be applied when a nonplanar orbit satisfying the completeness condition is used. Although Grangeat's inversion formula is mathematically different from Smith's one, they have similar overall structures to each other. The contribution of the present paper is two-fold. First, based on the derivation of Smith, the authors point out that Grangeat's inversion formula and Smith's one can be conveniently described using a single formula (the Smith-Grangeat inversion formula) that is in the form of space-variant filtering followed by cone-beam back projection. Furthermore, the resulting formula is reformulated for data acquisition systems with a planar detector to obtain a new reconstruction algorithm. Second, the authors make two significant modifications to the new algorithm to reduce artifacts and numerical errors encountered in direct implementation of the new algorithm. As for exactness of the new algorithm, the following fact can be stated. The algorithm based on Grangeat's intermediate function is exact for any complete orbit, whereas that based on Smith's intermediate function should be considered as an approximate inverse excepting the special case where almost every plane in 3D space meets the orbit. The validity of the new algorithm is demonstrated by simulation studies.

Weighted backprojection approach to cone beam 3D projection reconstruction for truncated spherical detection geometry

A new analytical three-dimensional cone beam reconstruction algorithm is presented for truncated spherical detection geometry. The basic idea of the proposed algorithm is the formation of spatially invariant 3D blurred back-projected volumetric image by the use of the weighted backprojection of cone beam projection data and subsequent 3D filtering using an acceptance angle dependent rho filter. The backprojection weighting function is calculated on the basis of each given geometrical condition, i.e. detection geometry or degree of truncation, position of cone beam apex, and backprojection point. The proposed algorithm is derived analytically and is computationally efficient. Performance of the algorithm is evaluated by the reconstruction of 3D volumetric images using simulated data from arbitrarily truncated spherical detector geometries.

A cone-beam filtered backprojection reconstruction algorithm for cardiac single photon emission computed tomography

A filtered backprojection reconstruction algorithm was developed for cardiac single photon emission computed tomography with cone-beam geometry. The algorithm reconstructs cone-beam projections collected from ;short scan' acquisitions of a detector traversing a noncircular planar orbit. Since the algorithm does not correct for photon attenuation, it is designed to reconstruct data collected over an angular range of slightly more than 180 degrees with the assumption that the range of angles is oriented so as not to acquire the highly attenuated posterior projections of emissions from cardiac radiopharmaceuticals. This sampling scheme is performed to minimize the attenuation artifacts that result from reconstructing posterior projections. From computer simulations, it is found that reconstruction of attenuated projections has a greater effect upon quantitation and image quality than any potential cone-beam reconstruction artifacts resulting from insufficient sampling of cone-beam projections. With nonattenuated projection data, cone beam reconstruction errors in the heart are shown to be small (errors of at most 2%).