Computational methods for the modelling and inversion of non-linear systems and its application to beam hardening correction in x-ray computed tomography reconstruction
In this project we will study different mathematical models that can be used to describe non-linear systems. We will focus particularly on the problem of model inversion, that is, for a known model, if you observe the output of the model, can you develop computational methods that allow you to compute the original model input? To help with this inversion, additional information on the input of the system will be assumed. This information will be formulated using sparse models, which have recently revolutionised several areas of science and engineering.
As a potential application, we will study the problem of x-ray tomography. Here the system we will model and invert describes the transmission of x-ray photons through an object (such as the human body). This mathematical model will describe non-linear effects found in most x-ray systems. By developing new computer algorithms to invert the non-linear model, we will design new methods that are able to produce better and more accurate images of the internal structure of an object. An important part of this work will be in the design of algorithms that can model and exploit additional image structure.