Assume again that *g = Pf* is known for .

We want to derive an inversion procedure similar to the one in section 2.1. With the backprojection

we have again

provided that . Again the convolutions on the left hand right hand side have different meanings. Explicitly this reads

which corresponds to (2.2). As in (2.2) we express the relationship in Fourier space, obtaining

see Colsher (1980). In order to get an inversion formula for *P* we have to determine *v* such that or , i.e.

A solution independent of is

where is the length of . For the spherical zone from section 3.4 with , , a constant with , Colsher computed explicitly. With be obtained

Filters such as the Colsher filter (3.10) do not have small support. This means that *g* in (3.8) has to be known in all of . Often *g* is only available
in part of (truncated projections). Let us choose

where from section (3.4) and is a horizontal unit vector. Since (3.11) is constant in the vertical direction, *v* is a -function in the vertical variable. Hence the
integral on the right hand side of (3.8) reduces to an integral over horizontal lines in , making it possible to handle truncated projections. Unfortunately,
(3.11) does not quite satisfy , i.e. it does not provide an exact inversion. Instead we only have

where , . This is close to if is small. In this case reconstruction from truncated projections is possible, at least approximately.

Thu Sep 10 10:51:17 MET DST 1998