Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 67 / 2 / f

In single-photon emission computed tomography, the unknown is an emitting source, rather than the absorption coefficient reconstructed by transmission computed tomography. Photons emitted at different points on the same ray are weighted by their probability of surviving the path to the detector. If the known attenuation is $a$ and the detector is at the positive end of the ray, the data are an attenuated Radon transformThe directional transport equation is $\mathbf{d}\cdot \nabla u+au=f$ with zero incoming intensity. An integrating factor derives this attenuation weight. The same complexification, boundary-value and jump strategy gives spectral reconstruction of an attenuated Radon transform when attenuation is known. The unattenuated inverse from part (d) applies directly only when $a=0$; for nonzero attenuation the weighted jump and inverse must be used. This distinction allows emission imaging of the source without confusing it with reconstruction of the absorbing medium.