Analytical solution to 3D SPECT reconstruction with non-uniform attenuation, scatter, and spatially-variant resolution variation for variable focal-length fan-beam collimators

In the past decades, analytical (non-iterative) methods have been extensively investigated and developed for the reconstruction of three-dimensional (3D) single-photon emission computed tomography (SPECT). However, it becomes possible only recently when the exact analytic non-uniform attenuation reconstruction algorithm was derived. Based on the explicit inversion formula for the attenuated Radon transform discovered by Novikov (2000), we extended the previous researches of inverting the attenuated Radon transform of parallel-beam collimation geometry to fan-beam and variable focal-length fan-beam (VFF) collimators and proposed an efficient, analytical solution to 3D SPECT reconstruction with VFF collimators, which compensates simultaneously for non-uniform attenuation, scatter, and spatially-variant or distance-dependent resolution variation (DDRV), as well as suppression of signal-dependent non-stationary Poisson noise. In this procedure, to avoid the reconstructed images being corrupted by the presence of severe noise, we apply a Karhune-Loève (K-L) domain adaptive Wiener filter, which accurately treats the non-stationary Poisson noise. The scatter is then removed by our scatter estimation method, which is based on the energy spectrum and modified from the triple-energy-window acquisition protocol. For the correction of DDRV, a distance-dependent deconvolution is adapted to provide a solution that realistically characterizes the resolution kernel in a real SPECT system. Finally image is reconstructed using our VFF non-uniform attenuation inversion formula.