International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 21, Pages 1305-1321

A general scheme for constructing inversion algorithms for cone beam CT

Alexander Katsevich

Department of Mathematics, University of Central Florida (UCF), Orlando 1364, FL, USA

Received 30 September 2002

Copyright © 2003 Alexander Katsevich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Given a rather general weight function n0, we derive a new cone beam transform inversion formula. The derivation is explicitly based on Grangeat's formula (1990) and the classical 3D Radon transform inversion. The new formula is theoretically exact and is represented by a 2D integral. We show that if the source trajectory C is complete in the sense of Tuy (1983) (and satisfies two other very mild assumptions), then substituting the simplest weight n01 gives a convolution-based FBP algorithm. However, this easy choice is not always optimal from the point of view of practical applications. The weight n01 works well for closed trajectories, but the resulting algorithm does not solve the long object problem if C is not closed. In the latter case one has to use the flexibility in choosing n0 and find the weight that gives an inversion formula with the desired properties. We show how this can be done for spiral CT. It turns out that the two inversion algorithms for spiral CT proposed earlier by the author are particular cases of the new formula. For general trajectories the choice of weight should be done on a case-by-case basis.