Linear diophantine equations for discrete tomography

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,..., M-1, with 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.