This formula inverts the ray transform
which comes up e.g. in 3D emission tomography (PET, Defrise et al. (1989)). If
is restricted to a plane, then we have simply the Radon transform in this plane, and we can
reconstruct f in that plane by any of the methods in the previous section. In practice
g = Pf is measured for
where
. In Orlov's formula
(Orlov (1976)),
is a spherical zone around the equator, i.e.
where ,
are the spherical coordinates of
and
. Then,
where is the Laplacian acting on x and
is the length of the intersection of
with the plane spanned by
. The first
formula of (3.6) is - up to
- a backprojection, while the second one a
convolution in
. Thus an implementation of (3.6) is again a filtered backprojection algorithm.
P can also be inverted by the Fourier transform. We have
where `` '' denotes the (n-1)-dimensional Fourier transform in
on the left hand side and the Fourier transform in
on the right hand side.
Assume that satisfies the Orlov condition: Every equatorial circle of
meets
. Note that the set
- the spherical zone - we used above in Orlov's formula satisfies this condition. From (3.7) it follows that f is uniquely determined by
for
under the Orlov condition. Namely if
is arbitrary, then Orlov's condition says that there exists
, and
is determined from (3.7).