Microlocal Analysis and Applications to Medical Imaging

This thesis is a collection of the three projects I have worked on at Purdue. The first is a paper on thermoacoustic tomography involving circular integrating detectors that was published in Inverse Problems and Imaging. Results from this paper include demonstrating that the measurement operators involved are Fourier integral operators, as well as proving microlocal uniqueness in certain cases, and also stability. The second paper, submitted to the Journal of Inverse and Ill-Posed Problems, is much more of an application of sampling theory in to the specific case of thermoacoustic tomography. Results from this paper include demonstrating resolution limits imposed by sampling rates, and showing that aliasing artifacts appear in predictable locations in an image when the measurement operator is under sampled in either the time variable or space variables. We also show an application of a basic anti aliasing scheme based on averaging of data. The last project moves slightly away from microlocal analysis and considers the uniqueness in medical imaging of the restricted Radon transform in even dimensions. This is the classical interior problem, and we show a characterization of the range of the Radon transform, and from this are able to obtain a characterization of the kernel of the restricted Radon transform. We include figures throughout to illustrate results.