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).

Thu Sep 10 10:51:17 MET DST 1998