Optimal recovery of a radiating source with multiple frequencies along one line

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation.

This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.