Frank Natterer
December 1994
A helical CT-scanner is a fan-beam scanner in which the patient is moved along the axis of the scanner with a constant speed as the x-ray source spins around the patient, allowing for 3D imaging [2]. We give the conditions on the scanning parameters (i.e. detector width, angular speed 
, translation speed v) for reliable reconstruction of a function of essential bandwidth 
(i.e. spatial resolution 
in all directions), and we describe an algorithm which actually achieves this resolution.
Assume that the patient is lying in a cylinder 
of radius 
around the axes (the 
axes) of the scanner, and that the source sits on a concentric circle of radius 
. Let f be a function which is supported in . We define
were 
is the first unit vector, and
In helical scanning the function g is sampled at
Here, 
is the detector spacing and 
the angle between two source positions. The condition that 
with an integer p makes the sampling lattice periodic in 
. We sketch the sampling points in the - -plane:
Thus the sampling points (2) are precisely the lattice
The reciprocal lattice is generated by the matrix
Determining the essential support of the Fourier transform of g and using the Petersen-Middleton sampling theorem would possibly lead to an efficient sampling scheme. In this note we pursue a less ambitious goal. We use the inversion formula for the 2D fan-beam transform ([1], p. 368)
Here, v is defined by
where 
is a filter factor vanishing outside [0,1], and the convolution in (4) with the function 
defined by
is with respect to the 2D variable 
. Convolving (4) in the variable with the filter
(or any other filter with bandwidth ) gives a reconstruction formula for helical CT, namely
deviates from the true f only in details of size 
. Applying the Petersen-Middleton sampling theorem to (5) as in [1] one obtains
Arranging the k-sum into complete cycles of length p and putting 
we obtain
For each , 
, d, L the k-sum is now a cyclic convolution of length p, making the evaluation amenable to FFT techniques. For each , only those values of k for which 
is close to contribute significantly. This means that only a few values of L have to be taken into account. Thus the evaluation of (7) for a fixed takes about the same number of operations as in 2D fan-beam tomography.
From [1] we know that the error in (6) is negligible provided that
the last condition being obvious. In view of (2) this translates into
If these conditions are met, no aliasing occurs in (6).
If one wants a different resolution in the x-direction, characterized by a bandwidth 
, one simply has to replace by in the last inequalities of (8) and (9).
