An analytical relationship between ideal and blurred sinograms and its implications in 2D SPECT

The effect of distance-dependent blurring is commonly formulated by studying how a point source in the image space is projected into the sinogram space. In the work reported here, however, the authors propose a novel perspective that focuses on tracing on how photon rays from the source is accepted by detectors in the presence of distance-dependent blurring. Based on this perspective, the concept of an angular response function of an imaging system is introduced and its relationship with the distance-dependent blurring function is established. The study reveals the direct relationship between the blurred and ideal sinograms, which is represented by an integration of the ideal sinogram along a known trajectory in the ideal sinogram space. The derived relationship provides the foundation to develop innovative methods that compensate for the effect of distance-dependent blurring in 2D SPECT. The authors also derived a 1D approximation of the direct relationship under realistic experimental conditions. This 1D approximated relationship describes a 1D convolution between the ideal sinogram and the modified angular response function. Therefore, ideal sinograms can be estimated from blurred sinograms by carrying out an effective 1D deconvolution with the use of various 1D deblurring methods