Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data

TL;DR

This paper develops a mathematical framework for simultaneously reconstructing radioactive sources and attenuation maps in SPECT imaging by leveraging ballistic and scattering data, proving local uniqueness, and proposing an iterative reconstruction algorithm.

Contribution

It introduces a nonlinear operator model for SPECT data, proves its invertibility under certain conditions, and develops a new iterative reconstruction method based on these theoretical insights.

Findings

Proved local invertibility of the linearized inverse problem.

Derived an explicit inverse operator for the linearized problem.

Proposed an iterative algorithm combining Neumann series and Newton-Raphson methods.

Abstract

In medical SPECT imaging, we seek to simultaneously obtain the internal radioactive sources and the attenuation map using not only ballistic measurements but also first order scattering measurements. The problem is modeled using the radiative transfer equation by means of an explicit nonlinear operator that gives the ballistic and scattering measurements as a function of the radioactive source and attenuation distributions. First, by differentiating this nonlinear operator we obtain a linearized inverse problem. Then, under regularity hypothesis for the source distribution and attenuation map and considering small attenuations, we rigorously prove that the linear operator is invertible and we compute its inverse explicitly. This allows to prove local uniqueness for the nonlinear inverse problem. Finally, using the previous inversion result for the linear operator, we propose a new type of iterative algorithm for simultaneous source and attenuation recovery for SPECT based on Neumann series and a Newton-Raphson algorithm.