By Simon Göppel

1. Computed Tomography

The goal of x-ray-tomography (greek τoµoς = slice) is to reconstruct the interior of an unknown two dimensional object from a collection of measure- ments, which are given as one dimensional projections. In order to obtain a two dimensional image from lower dimensional data, these measurements are taken from many different views. A computed tomography (CT) scanner does this by rotating a source-detector pair around the analysed object, see Figure 1.1. The x-rays can be assumed to travel along straight lines (in- dicated by the dashed arrows), emitted by the source and measured at the detector.

Fig. 1.1. Illustration of a computed tomography scanner

The underlying imaging principle is based on the fact that the object attenuates some of the energy of the x-rays. The strength of attenuation depends on the objects interior structure and materials, hence, carries infor- mation that can be used to create a two dimensional representation. The intensity of the x-ray, measured at the detector, is given by Beer’s law

where I₀ denotes the initial intensity emitted at the source, L models the trajectory of the x-ray and f describes the absorption coefficient at the given position. Using simple algebra, we can restate Beer’s law as

In particular, knowing the initial and measured intensity I₀, I, recovering the attenuation coefficient, or in other words, reconstructing an image of the interior of the object, is the same as solving equation (1.1) for f .

2. Random Transform and Invere Problems

The mathematical theory behind solving equation (1.1) is well developed if the data is available for all possible lines L. In fact, already in 1917, Austrian mathematician Johann Radon published a pioneering paper with the  title  ”Ü ber  die  Bestimmung  von  Funktionen  durch  ihre  Integralwerte längs gewisser Mannigfaltigkeiten”, where he gave an exact formula to recover f . To this day, the Radon transform, denoted by R, suits as the mathematical model for CT.

Thus, reconstructing the image f from given measurements, can also be interpreted as an operator equation of the form

Here, g denotes the data produced by the CT scanner (left-hand side of (1.1)), η models the noise or perturbation of the measurements and Rf describes the measurement process (right hand side of (1.1)). To obtain f , one has to calculate the inverse of R and apply it on both sides of the equation.

However, the equation (2.1) relates to a so-called ill-posed inverse prob- lem. This means that small measurement errors, can lead to severe errors in the reconstructions. This is not only the case for CT, but also for a variety of problems, that arise in almost all scientific areas. In general, the research goal of inverse problems is concerned with re-establishing well-posdness of equa- tion (2.1) again, that is to make the reconstruction process stable against data perturbations.

This can be done, for example, by incorporating as much information as possible from the signal f into a suitable reconstruction algorithm. Math- ematically, this leads to the so-called regularization theory, where the goal is to find suitable regularization methods that are well adapted to a given problem of the form (2.1). In my work, I try to define novel regularization strategies and frameworks. This involves the study of related theories as well as programming and numerical simulations. Besides the adaptation and improvement of classical regularization methods, this also includes the inte- gration of state-of-the-art machine learning tools alongside theoretical and numerical analysis.