Computed Tomography and Sampling

Oregon State University, Corvallis OR

Investigators

Abstract

This project involves research in computed tomography, advanced Shannon sampling theory, and applications of sampling theory in tomography. It combines theoretical investigations, numerical analysis, and computational experiments with real world data. Computed tomography is a technique which produces images of the interiors of opaque objects. It is widely used in applications ranging from diagnostic radiology to research in quantum optics. Mathematically it requires the reconstruction of a certain density function from 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" or "local tomography"). Shannon sampling theory plays 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. The project's research in tomography involves interdisciplinary collaboration and is inspired by the question "What features of the density function can be stably recovered from a given collection of its line integrals?". Specific goals include the identification and analysis of optimal sampling schemes in three-dimensional tomography; the analysis of reconstruction algorithms for three-dimensional local tomography with sources on a curve; further development and numerical analysis of high-resolution reconstruction algorithms in two dimensions, and development of a software package for region-of-interest tomography. The research in sampling theory is stimulated by applications in tomography and involves the development and application of new sampling theorems for non-equidistant data; estimates for the aliasing error in various settings; and further exploration of a generalization of the sampling concept in the framework of locally compact abelian groups which provides a unified view of a number of diverse applications.