Multi-bang Regularization and Applications

Abstract

Inference of the internal structure of an object from passive radiation imaging has many applications in modern day life, ranging from Medical Imaging to Nuclear Security. In this thesis we focus on the joint reconstruction of attenuation /a/ and radiation source density /f/ from the Attenuated Radon Transform (AtRT) /R_{a}f/ which models Single-Photon Emission Computed Tomography (SPECT) data. Joint inversion in the general case is known to be impossible and we instead consider the setting were /a/ takes only finitely many values, which we refer to as ``multi-bang'', and /f/ is once differentiable with compact support. In this setting we are able to characterise singularities appearing in the AtRT. With constraints on the support of /f/ in relation to the support of /a/ and mild conditions on the boundaries of /a/, we are able to show unique recovery of the sets on which /a/ is constant. When the sets making up /a/ are nested convex sets, referred to here as "nicely multi-bang", we show unique recovery of /a/ and /f/. It is also possible to obtain partial results for the more general case ranging from the complete determination in special cases to situations where we can at least determine /a/ in certain sets. We also propose a numerical algorithm to jointly compute /a/ and /f/ from /R_{a}f/ based on a weakly-convex regularizer, referred to as a multi-bang regularizer. Various numerical examples are given to show that the algorithm performs well on synthetic examples.

Date of Award	1 Aug 2021
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Oliver Dorn (Supervisor) & Sean Holman (Supervisor)