MATJ5111 MA2: Introduction to the mathematics of X-ray imaging: X-ray transforms (JSS31) (1 cr)

Computerized Tomography, used every day in hospitals to produce internal images of patients, is based on the inversion of the Radon transform. In two spatial dimensions, this transform maps a function to the collection of its integrals along all lines through the plane. More general 'X-ray' transforms can be defined by integrating along different families of curves, with applications to, e.g., seismology. These transforms are examples of integral-geometric operators and their inversion is what constitutes an inverse problem. The task of inverting these transforms can be approached by addressing analytical questions such as injectivity and stability, and by attempting to find inversion formulas.
In this mini-course, we will first discuss the analytical framework behind the study of general inverse problems. We will then focus on studying three prototypes of integral-geometric inverse problems whose study can be made rather explicit via Fourier-based techniques: the Funk transform on the sphere, the Radon transform on the plane, and the X-ray transform on the unit disk. Time permitting, numerical examples and/or a discussion on possible generalizations will be given.

Students familiarize themselves with the model behind X-ray tomography. They also become familiar with the prototypical questions related to the analysis of inverse problems (injectivity, stability, inversion) and the methods to address these questions in the context of integral-geometric problems, gaining exposure to a mix of functional-analytic, Fourier-analytic and geometric concepts along the way.

Basic functional analysis.

Bibliography: Lecture notes will be provided.

Other available resources:
- F. Natterer, The mathematics of Computerized Tomography, Society for Industrial and Applied Mathematics, 2001.
- G. Paternain, M. Salo and G. Uhlmann, Geometric Inverse Problems with emphasis on two dimensions, 2021.
- J. Ilmavirta and F. Monard, Integral geometry on manifolds with boundary and applications, The Radon
Transform: the first 100 years and beyond. Radon series on computations and applied mathematics 22 (2019). Editors: R. Ramlau, O. Scherzer.
- J. Ilmavirta, Analysis and X-ray tomography, lecture notes (2017), arXiv:1711.06557.