Theory for image reconstruction from divergent-beam projections in SPECT

This work addresses image reconstruction from divergent-beam projections in 2D and 3D SPECT. Assuming that the attenuation map is constant in a convex region Omega that includes all activity sources, we show that the problem of reconstruction from divergent-beam projections can be reduced to solving a 1D integral equation, and this for any point that lie on an R-line, which is a segment of line that connects two vertex points. We call vertex point, the converging point of the collimator, and we call vertex path, the path followed by the vertex point during data acquisition. The 1D integral equation turns out to be one we were able to analyze in details in another work. In the context of the present work, this analysis leads to the following data sufficiency condition: accurate reconstruction is possible at any point that lies on an R-line provided the intersection of Omega with this R-line is fully visible at all time in the projection data while the vertex point moves from one extremity of the R-line to the other along the vertex path. For fan-beam SPECT, this result implies that accurate reconstruction of a region-of-interest is possible from projections that are collected on less than a short-scan and may furthermore be each truncated. For cone-beam SPECT, this result implies for example that with a helical data acquisition, accurate reconstruction is possible anywhere inside the helix cylinder when (i) there is no transaxial truncation, and (ii) the pitch of the helix is small enough so that the gamma camera captures the lines that are just within the Tam-Danielsson window. Our theoretical result is supported by 2D and 3D experimental results obtained from computer-simulated data.