This project is to investigate questions of uniqueness, stability, consistency, asymptotic behavior and analytic and numerical inversion of operators that arise in mathematical models of several inverse problems. While the research is motivated by reconstruction problems in medical radiology, the operators under study (Radon transforms, x-ray transforms and generalization) are of importance in numerous other applications. The main topics for investigation are: the further development and testing of a computationally efficient cone beam reconstruction algorithm with sources on a circle; the study of stability effects of data truncation in the algorithm and the cone beam reconstruction problem in general, and a singular value analysis of operators arising in this problems; a group theoretic study of filtered backprojection inversion and approximate inversion formulas in, and applications of wavelet theory to, integral geometric inverse problems. Successful completion of the project should lead to a computationally efficient algorithm for cone beam reconstruction, a mathematical understanding of the stability of the algorithm and the inherent stability or instability of the problem. Other benefits include a theoretical understanding of the underpinnings of filtered backprojection inversion formulas and the structure of operators arising in integral geometric inverse problems.
