A general scheme for constructing inversion algorithms for cone beam CT

Abstract

Given a rather general weight functionn0, 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 trajectoryCis complete in the sense of Tuy (1983) (and satisfies two other very mild assumptions), then substituting the simplest weightn0≡1gives a convolution-based FBP algorithm. However, this easy choice is not always optimal from the point of view of practical applications. The weightn0≡1works well for closed trajectories, but the resulting algorithm does not solve the long object problem ifCis not closed. In the latter case one has to use the flexibility in choosingn0and 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.