Emission tomography with a multi-bang assumption on attenuation

Abstract

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, or referred to as attenuation correction in SPECT. Assuming that a takes only finitely many values and f ∈ C 1 c (R 2 ) we are able to characterise singularities appearing in the Attenuated Radon Transform Raf , which models emission tomography data. Using this characterisation we prove that both a and f can be determined in some circumstances. 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.