This project involves research in computed tomography, generalized Shannon sampling theory, and applications of sampling theory in tomography. Computed tomography is an imaging technique which reconstructs a density function from a large number of its line integrals. Ordinary tomography is not local: reconstruction at a point requires integrals over lines far from that point. For a number of applications it is highly desirable if only integrals over lines intersecting some region of interest need to be used ("region-of-interest tomography"). A special case is local tomography where the region of interest may be arbitrarily small. Local tomographic reconstructions recover some but not all properties of the original density function. Sampling theorems provide exact interpolation formulas for bandlimited functions and play a fundamental role in signal processing. In tomography sampling theorems are used to identify efficient data collection schemes allowing for maximum resolution in the reconstructed image, as well as for error analysis of reconstruction algorithms. If, as is the case in tomography and other applications, sampling theorems are applied to non-bandlimited functions, interpolation is no longer exact and a so-called aliasing error occurs. The usefulness of sampling theorems for such functions depends on the availability of good estimates for this error.
The goals of this project are development and numerical analysis of high-resolution reconstruction algorithms and development of a software package for region-of-interest tomography including local tomography; investigation of optimal sampling in three-dimensional local tomography; derivation of new error estimates for the aliasing error; and exploration of a generalization of the sampling concept in the framework of locally compact abelian groups which provides a unified view of diverse applications such as non-equidistant sampling, numerical integration, multisensor deconvolution, and filter banks.
