Corrections for Physical Factors - Correction Techniques in Emission Tomography - page 93

In 2D PET, this convolution is a simple 1D operation on the projection
data while in 3D PET it is a 2D convolution on the projection planagrams [6].
The main task now is to solve the equation for g u . This can be done by direct
deconvolution approaches [29] or by convolution-subtraction methods [6]. The
former makes use of the convolution theorem of Fourier transformations
g u = F 1

F(g 0 )
F ( + k

; (5.23)

· f)

with the Fourier transformation F and the Dirac distribution , while the
latter uses an iterative approach to determine g u :

g (1 u = g 0 k

· (g 0 f) (5.24)

g (n1 u f ;

g (n u = g 0 k

· (5.25)

where g (n u denotes the nth iterative estimate of g u . Both scatter fraction k
and scatter function f are usually modeled according to phantom measure-
ments. A monoexponential function of the form f(x) / exp(jxj) is often
chosen in PET. Although this approach does not take spatially inhomoge-
neous scattering into account, it has been shown that it leads to satisfying
results in removing scattered events in 3D PET of phantom and human brain
studies [90]. More accurate scatter models in both SPECT and PET take
non-uniform scattering (i.e., spatially variant scatter functions and fractions)
into account by using the acquired transmission data [59] [66]. Alternatively,
the convolution-subtraction algorithm can also be applied to the PET data
after reconstruction. This is done by independent reconstruction of both the
measured data g 0 and the determined scatter distribution g s and then finally
subtracting the scatter image from the PET image [54].
Another way of using simple analytical functions for scatter correction of
PET data is realized in tail-fitting methods. The basic idea here is the fact
that measured coincidence events on lines that do not pass the scanned object
must be scattered events. (Note that these methods do not work in SPECT,
as scattered events appear to be confined to the scanned object in SPECT.)
The measured data outside the object, the tail, is fitted with an analytical
function such as a Gaussian which is then interpolated inside the object to
get an estimate of the overall scatter distribution (Figure 5.21). This inter-
polation is legitimate due to the assumption that the scatter distribution is
characterized as a low frequency function that does not strongly depend on
high frequency distributions in the radiotracer distribution. The determined
scatter distribution is then subtracted from the measured data. This method
has been successfully implemented in human brain PET [22]. One of its advan-
tages is that it takes scatter from outside the field of view into account which
is a known problem especially in 3D PET. However, this method may fail in
cases with asymmetric scatter distributions caused by non-uniform objects.