Analytical inversion formula for attenuated fan-beam projections

In single photon emission computed tomography (SPECT), photon attenuation within the body is a major factor contributing to the quantitative inaccuracy in measuring the in vivo distribution of radioactivity. Usually the attenuation of the body is not uniform, but for brain imaging, it can be a good approximation to assume that the attenuation is uniformly distributed. For 2D parallel-beam geometry, an exact convolution backprojection algorithm to reconstruct image from attenuated Radon transform with constant attenuation had been developed by Tretiak and Metz (1980). The algorithm can be modified for attenuated fan-beam projections. Unlike the attenuated parallel-beam projections, the filter for attenuated fan-beam projections is no longer spatially invariant, instead, it is a space-variant filter. The algorithm with this spatially variant filter will take more computation time than the algorithm with convolution, but is an exact algorithm. This algorithm has been implemented and simulated using a mathematical phantom. Compared with parallel-beam reconstructions, fan-beam reconstructions have the same image quality