Geodesic tomography problems on Riemannian manifolds

TL;DR

This dissertation investigates integral geometric inverse problems on Riemannian manifolds, establishing conditions for unique and stable function recovery from geodesic ray transforms, and introduces a new numerical model for computed tomography.

Contribution

It provides new conditions for uniqueness and stability in geodesic tomography and develops a novel numerical model for computed tomography imaging.

Findings

Conditions for unique function determination from geodesic data

Stability estimates for inverse problems on Riemannian manifolds

A new numerical model for computed tomography imaging

Abstract

This PhD dissertation is concerned with integral geometric inverse problems. The geodesic ray transform is an operator that encodes the line integrals of a function along geodesics. The dissertation establishes many conditions when such information determines a function uniquely and stably. A new numerical model for computed tomography imaging is created as a part of the dissertation. The introductory part of the dissertation contains an introduction to inverse problems and mathematical models associated to computed tomography. The main focus is in definitions of integral geometry problems, survey of the related literature, and introducing the main results of the dissertation. A list of important open problems in integral geometry is given. The four articles of the dissertation (arXiv:1705.10126, arXiv:1901.03525, arXiv:1906.05046, arXiv:1909.00495) are now published in various journals and the content of the document has not been updated since November 11, 2019. These four articles are omitted from this document which contains only the introductory part. Extended abstracts are given in the document in English and Finnish.