X-ray transform and 3D Radon transform for ellipsoids and tetrahedra

Accurate simulation of X-ray transforms of representative objects plays an important role in the evaluation and improvement of CT reconstruction algorithms. In this paper, we formulate the X-ray transform and 3D Radon transform for ellipsoids and tetrahedra, and verify the resulting formulas by numerical simulation. Here the ellipsoids and tetrahedra may be arbitrarily positioned and rotated. Linearity of the first derivative of radon transform is observed. Our results serve as a benchmark for development of various cone-beam algorithms that work through the Radon space, such as Grangeat-type algorithms.