Analytical fan-beam and cone-beam reconstruction algorithms with uniform attenuation correction

In this paper, we developed an analytical fan-beam reconstruction algorithm that compensates for uniform attenuation in SPECT. The new fan-beam algorithm is in the form of backprojection first then filtering, and is mathematically exact. The algorithm is based on three components. The first one is the established generalized central slice theorem, which relates the 1D Fourier transform of a set of arbitrary data and the 2D Fourier transform of the backprojected image. The second one is the fact that the backprojection of the fan-beam measurements is identical to the backprojection of the parallel measurements of the same object with the same attenuator. The third one is the stable analytical reconstruction algorithm for uniformly attenuated Radon data, developed by Metz and Pan. The fan-beam algorithm is then extended into a cone-beam reconstruction algorithm, where the orbit of the focal-point of the cone-beam imaging geometry is a circle. This orbit geometry does not satisfy Tuy's condition and the obtained cone-beam algorithm is an approximation algorithm. In the cone-beam algorithm, the cone-beam data are first backprojected into the 3D image volume, then a slice-by-slice filtering is performed. This slice-by-slice filtering procedure is identical to that of the fan-beam algorithm. Both the fan beam and cone-beam algorithms are efficient and the same technique can be used to develop efficient analytical algorithms for non-uniform attenuators.