Linear diophantine equations for discrete tomography

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.