It is well known (see e.g. Kruse (1989)) that the data in the standard parallel geometry is redundant:
If p is even, then one can omit each with
odd
without impairing the resolution. Deriving algorithms which use only the remaining
``interlaced'' data (i.e. those
for which
even) is fairly subtle. What happens is the following. If in the
sum in (2.4) every
second term is droped, the sum no longer approximates the corresponding s
integral in (2.2). Miraculously the large quadrature error cancels when
the j sum in (2.4) is computed. This means that success depends entirely on a subtle interplay between different directions. This interplay is disrupted by the
interpolation procedure in step 2 of Algorithm 1. There are two ways out. The first
one is to avoid interpolation alltogether by using circular harmonic algorithms, see
section 5. The second one is to make the interpolation more accurate, for instance by
oversampling. This leads to an algorithm which has the structure of a filtered
backprojection algorithm.
Algorithm 2 (Filtered backprojection algorithm for parallel interlaced geometry.)
where k = k(j,x), are determined by
Note that the difference between this algorithm and Algorithm 1 is that it needs only one half of the data but produces the same image quality. We study the various assumptions underlying this algorithm.