Image Reconstruction from Incomplete Radon Data and Generalized Principal Component Analysis

Abstract

Image reconstruction in various types of tomography requires inversion of the Radon transform and its generalizations. While there are many stable and robust algorithms for such inversions from reasonably well sampled data, most of these algorithms fail when applied to limited view data. In the dissertation we develop a new method of stable reconstruction from limited view data for functions, whose support is a union of finitely many circles. Such images, among other things, are good approximations of tomograms of certain types of tumors in lungs. Our method is based on a modified version of GPCA (General Principle Component Analysis) and some results from algebraic geometry.