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 n0≡1 gives a convolution-based FBP
algorithm. However, this easy choice is not always optimal from
the point of view of practical applications. The weight
n0≡1 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.