Inverse Problems in Tomography and Fast Methods for Singular Convolutions

Abstract: There are two, partially interlaced, themes treated in this thesis; inverse problems of tomographic type and fast and accurate methods for the application of convolution operators. Regarding the first theme, the inverse problem of Doppler tomography is considered and the Doppler moment transform is introduced for that purpose. By investigating the properties of the transform, we prove results regarding uniqueness and develop a numerical method for reconstruction. Continuing on the tomography track, we turn focus to X-ray tomography and construct a fast numerical method for the inversion of the Radon transform. Our method is of filtered back-projection type, the most commonly used in practice, but has one order lower time complexity than the standard methods. Moving on to the field of diffraction tomography, we develop a fast method for solving the forward scattering problem in inhomogeneous media. To this end, we employ the methods for fast and accurate application of singular convolution operators that constitute the second main theme of the thesis. First, we consider the problem of fast Gauss transform with complex parameters, and then develop further some of the results to construct a continuous framework for fast application of convolution operators.