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 $f\in C_c^1({\mathbb{R}}^2)$ we are able to characterise singularities appearing in the Attenuated Radon Transform $R_{a}f$, 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 $R_{a}f$ 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.

23 pages, 7 figures

References in corpus (1)

• On the Convergence of the Iterative Shrinkage/Thresholding Algorithm With a Weakly Convex Penalty