Simultaneous recovery of attenuation and source density in SPECT

TL;DR

This paper proves that under certain conditions, the attenuated Radon transform uniquely determines piecewise constant attenuation and piecewise smooth source density in SPECT, with numerical evidence suggesting possible uniqueness even when conditions fail.

Contribution

The paper establishes theoretical uniqueness results for simultaneous recovery of attenuation and source density in SPECT under a specific non-cancellation condition, and provides numerical examples exploring cases where this condition fails.

Findings

Unique determination of attenuation and source density under non-cancellation condition

Numerical evidence of possible uniqueness without the non-cancellation condition

Piecewise constant and piecewise smooth functions can be reconstructed in SPECT

Abstract

We show that under a certain non-cancellation condition the attenuated Radon transform uniquely determines piecewise constant attenuation $a$ and piecewise $C^2$ source density $f$ with jumps over real analytic boundaries possibly having corners. We also look at numerical examples in which the non-cancellation condition fails and show that unique reconstruction of multi-bang $a$ and $f$ is still appears to be possible although not yet explained by theoretical results.