Kernel-Based Generalized Interpolation and its Application to Computerized Tomography

• K. Albrecht

2024, Doctoral Thesis: 178 p.

• Final published version

Abstract

Throughout recent decades, positive definite kernel functions have turned out to be powerful and flexible approximation tools for several mathematical problems and their associated real-world applications. Besides the interpolation of Lagrangian data, kernel functions are well-suited to solve more general interpolation problems concerning the evaluation of arbitrary functionals, also known as generalized interpolation. Although the treatment of the generalized case is straightforward in many aspects, it has not gained the same attention as the standard interpolation case yet. So far, the research on this topic has mainly focused on the solution of partial differential equations, and further applications such as medical imaging are rather exotic.

In 2018, the authors De Marchi, Iske and Santin proposed the application of generalized interpolation to the field of computerized tomography in combination with weighted kernel functions. Inspired by this work, the objective of this thesis is to further elaborate the concept of generalized interpolation and investigate its utility for the reconstruction of images from scattered Radon data. To this end, we derive an extensive framework for treating generalized interpolation problems in the first part of this thesis, which includes data-dependent orthonormal systems, greedy data selection algorithms and suitable regularization methods. We provide convergence results under mild assumptions on the considered kernel and translate several results from standard Lagrangian interpolation to the generalized case.

The derived framework is then applied to the problem of computerized tomography in the second part, where we derive useful properties of weighted kernel functions. By choosing suitable kernels, we can guarantee the well-posedness of the reconstruction method and therefore make use of our general theoretical results from the first part. Moreover, we provide theoretical and numerical comparisons to other well-established reconstruction methods to demonstrate the advantages of kernel-based generalized interpolation in the field of computerized tomography.