An exact inversion formula written in the form of shift-variant filtered-backprojection (FBP) is given for reconstruction from cone-beam data taken from any orbit satisfying H.K. Tuy's (1983) sufficiency conditions. The method is based on a result of P. Grangeat (1987), involving the derivative of the three-dimensional (3D) Radon transform, but unlike Grangeat's algorithm, no 3D rebinning step is required. Data redundancy, which occurs when several cone-beam projections supply the same values in the Radon domain, is handled using an elegant weighting function and without discarding data. The algorithm is expressed in a convenient cone-beam detector reference frame, and a specific example for the case of a dual orthogonal circular orbit is presented. When the method is applied to a single circular orbit (even though Tuy's condition is not satisfied), it is shown to be equivalent to the well-known algorithm of L.A. Feldkamp et al. (1984).
The current trend in attenuation correction for single photon emission computed tomography (SPECT) is to measure and reconstruct the attenuation coefficient map using a transmission scan, performed either sequentially or simultaneously with the emission scan. This approach requires dedicated hardware and increases the cost (and in some cases the scanning time) required to produce a clinical SPECT image. Furthermore, if short focal-length fan-beam collimators are used for transmission imaging, the projection data may be truncated, leading to errors in the attenuation coefficient map. Our goal is to obtain information about the attenuation distribution from only the measured emission data by exploiting the fact that only certain attenuation distributions are consistent with a given emission dataset. Ultimately this consistency information will either be used directly to compensate for attenuation or combined with the incomplete information from fan-beam transmission measurements to produce a more accurate attenuation coefficient map. In this manuscript the consistency conditions (which relate the measured SPECT data to the sinogram of the attenuation distribution) are used to find the uniform elliptical attenuation object which is most consistent with the measured emission data. This object is then used for attenuation correction during the reconstruction of the emission data. The method is tested using both simulated and experimentally acquired data from uniformly and nonuniformly attenuating objects. The results show that, for uniform elliptical attenuators, the consistency conditions of the SPECT data can be used to produce an accurate estimate of the attenuation map without performing any transmission measurements. The results also show that, in certain circumstances, the consistency conditions can prove useful for attenuation compensation with nonuniform attenuators.
Three-dimensional medical image reconstruction for both transmission and emission tomography has traditionally decomposed the problem into a set of two-dimensional reconstructions on parallel transverse sections. There is, however, increasing interest in reconstructing projection data directly in three dimensions. For emission tomography in particular, such a reconstruction procedure would clearly make more efficient use of the available photon flux. In the past few years, a number of authors have studied the problems associated with full three-dimensional reconstruction, especially in the case of positron tomography where three-dimensional reconstruction is likely to offer the greatest benefits. While most approaches follow that of filtered backprojection, the relationship between the various filters that have been proposed is far from evident. This paper clarifies this relationship by analysing and generalising the different classes of published filters and establishes the properties and characteristics of a general solution to the three-dimensional reconstruction problem. Some guidelines are suggested for the choice of an appropriate filter in a given situation.
Cone-beam data acquired with a vertex path satisfying the data sufficiency condition of Tuy can be reconstructed using exact filtered backprojection algorithms. These algorithms are based on the application to each cone-beam projection of a two-dimensional (2-D) filter that is nonstationary, and therefore more complex than the one-dimensional (1-D) ramp filter used in the approximate algorithm of Feldkamp, Davis, and Kress (1984) (FDK). We determine in this paper the general conditions under which the 2-D nonstationary filter reduces to a 2-D stationary filter, and also give the explicit expression of the corresponding convolution kernel. Using this result and the redundancy of the cone-beam data, a composite algorithm is derived for the class of vertex paths that consist of one circle and some complementary subpath designed to guarantee data sufficiency. In this algorithm the projections corresponding to vertex points along the circle are filtered using a 2-D stationary filter, whereas the other projections are handled with a 2-D nonstationary filter. The composite algorithm generalizes the method proposed by Kudo and Saito (1990), in which the circle data are processed with a 1-D ramp filter as in the FDK algorithm. The advantage of the 2-D filter introduced in this paper is to guarantee that the filtered cone-beam projections do not contain singularities in smooth regions of the object. Tests of the composite algorithm on simulated data are presented.
We investigate the way data are used in the algorithm proposed by Kudo and Saito for the exact reconstruction of long objects from axially truncated cone-beam projections. Specifically, we show that the algorithm wastes a large part of the data. To overcome the problem, we propose to use a vertex path consisting of two crossing ellipses, for which we devised a new reconstruction algorithm, called the cross algorithm, which does not waste data and is still suitable to exactly handle axial truncation. Results of reconstruction are presented on simulated data and real data from an experimental scanner.
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