Investigations of Some Direct and Inverse Problems in X-Ray In-Line Phase-Contrast Imaging and Tomography

Abstract

In the present thesis several direct and inverse problems of X-ray in-line phase-contrast imaging and computed tomography are studied. A general method for finding the fundamental solution of the Helmholtz equation subject to Sommerfeld radiation conditions is developed. Unlike the established techniques, this method provides all solutions to the Helmholtz equation before selecting the one that satisfies the chosen valid boundary conditions. Sufficient conditions for the validity of Teague's method for solving the Transport of Intensity Equation are derived and an example of a solution, which cannot be obtained using this method, is provided. Teague's method is also applied to tomography for the reconstruction of the three-dimensional refractive index distribution in a generic sample from in-line X-ray projections. The proposed solution simplifies and stabilises the reconstruction process. A formula is derived for the single-step reconstruction of a newly introduced auxiliary function. This function contains information about both the absorption index and the refractive index decrement. The reconstruction is obtained directly from the intensity measurements, without the intermediate step of phase retrieval for each illumination angle. A precise relationship is established between the newly introduced function and the complex refractive index distribution. The physical meaning of this function is examined for phase objects and for generic objects with slowly varying distributions of absorption index. Some examples of possible applications of our results are discussed in Conclusions.