Here, the detector positions within a fan with vertex *b* are evenly spaced on the
line perpendicular to *b*. We need the explicit form of the inversion formula mainly
for the derivation of the FDK algorithm in 3D cone beam tomography in the next
section. With *g* the function in (2.8), the sampled data is

The coordinates , *y* are related to the parallel coordinates , *s*
in the representation of the rays by

Hence,

Substituting , *y* for , *s* in (2.2) leads to

where (2.12) has to be inserted for , *s*. As in the standard fan beam
case a direct implementation of this formula results in an algorithm whose complexity is not competitive. Again we can circumvent this problem by exploiting the homogeneity properties
of *v*. A lengthy but elementary computation shows that

From (2.11) it follows that

yielding

As in the standard fan beam case we make the approximation . Again this is justified if , e.g. . Then,

where . Defining

this can be written as

The implementation of (2.13), (2.14) can now be done exactly as in the
standard case, leading to a filtered backprojection algorithm which needs O*(p)* operations
for each reconstruction point *x*.

Thu Sep 10 10:51:17 MET DST 1998