Combined source and attenuation reconstructions in SPECT

We consider the simultaneous reconstruction of the absorption coefficient and the source term in a linear transport equation from available boundary measurements. This problem finds applications in SPECT, a medical imaging modality. When the absorption coefficient is known, recently derived inversion formulas for the attenuated Radon transform can be used to reconstruct the source term. Moreover, the measurements needs to satisfy some compatibility conditions, which fully characterize the range of the attenuated Radon transform. In this paper, we explore this compatibility condition to obtain information about the absorption coefficient. We consider a linearization of the compatibility condition and show that the absorption term is uniquely determined, partially determined, or fully undetermined, depending on the structure of the source term.