Linear diophantine equations for discrete tomography

Issue title: Special Issue on Clinical Applications of Modern Imaging

Article type: Research Article

Authors: Ye, Yangbo | Wang, Ge | Zhu, Jiehua

Affiliations: Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA | Department of Radiology, The University of Iowa, Iowa City, IA 52242, USA

Abstract: In this report, we present a number-theory-based approach for discrete tomography (DT),which is based on parallel projections of rational slopes. Using a well-controlled geometry of X-ray beams, we obtain a system of linear equations with integer coefficients. Assuming that the range of pixel values is a(i,j)=0,1<FORMULA>, &mldr;, <FORMULA>M-1<FORMULA>, with <FORMULA>M being a prime number, we reduce the equations modulo M. To invert the linear system, each algorithmic step only needs log^2_2 M bit operations. In the case of a small M, we have a greatly reduced computational complexity, relative to the conventional DT algorithms, which require log^2_2 N bit operations for a real number solution with a precision of 1/N. We also report computer simulation results to support our analytic conclusions.

Keywords: computed tomography (CT), discrete tomography (DT), number theory, Diophantine equations, computational complexity

Journal: Journal of X-Ray Science and Technology, vol. 10, no. 1-2, pp. 59-66, 2001