SPECT with a multi-bang assumption on attenuation

We consider the problem of joint reconstruction of both attenuation a and source density f in emission tomography in two dimensions. This is sometimes called the Single Photon Emission Computed Tomography (SPECT) identification problem. Assuming that a takes only finitely many values and f is once differentiable with compact support, we are able to characterise singularities appearing in the Attenuated Radon Transform Raf, which models SPECT data. Using this characterisation we prove that both a and f can be determined in some circumstances from Raf. We also propose a numerical algorithm to jointly compute a and f from Raf based on a weakly convex regularizer when a only takes values from a known finite list, and show that this algorithm performs well on some synthetic examples.