PET Image Reconstruction - Correction Techniques in Emission Tomography - page 46

Also note that the algorithm is different from a competing approach which
is tempting as a simple motivation: for each measured line L, write the equa-
tion

Z

f(x)ds = 1;

L

discretize and apply EM to the resulting system. This would result in the same
algorithm as (3.4), with the exception that the normalization matrix would
have to be chosen as A LM as well. The simple example of a delta source at
the origin shows that this makes no sense.
At first glance, this approach (choosing a very fine grid of lines) seems
to break the interpretation of g as an approximation to the X-ray transform,
since we have only 1s and 0s in g. However, this is not true: assume that in an
experiment classical EM is applied and all lines measured are members of S,
meaning that no precision is lost in rebinning. Note that this is the case with
classical crystal sensor elements where data is naturally prediscretized since
only a finite number of directions can be measured. Suppose further that line
L 2 S has been awarded g L events. In EM, we need to compute
A t g

Af k :

We construct a matrix A EM which consists of the rows of A the row belonging
to line L in A is written g L times in A EM . Then the update above just reads

1
A EM f k :

A t EM

Since A EM differs from A LM only in the order of rows, in this case, EM list
mode is exactly the same as classical EM. Note that this is valid only for
EM, not for OSEM-like methods. In OSEM, the arrangement of the matrix is
crucial, so it is not the same thing.
Since EM list mode is just a special case of EM, all convergence acceleration
methods like OSEM, optimal choice of convergence parameters and so on can
and must be applied.
However, there is a serious drawback, which of course is computation time.
The size of system matrix A is much bigger than in the classical rebinned
system; we get one equation per measured event. This is in part accounted
for by an optimal randomization of events: Not only are the single rebinned
equations written in a random order, but from the rebinning point of view
even the equations are split into single equations with 1 on the right hand
side and randomized. So a major reduction of number of iterations can be
expected and is usually seen.
List mode statistics have been investigated thoroughly in papers by Parra
and Barrett [12].