Physics and Radiobiology of Nuclear Medicine

where Cjis the counts (activity) in the jth pixel and aijis the probability that
an emission from pixel jis recorded in the ith projection. The weight,aij,is
equal to the fraction of activity in the jth pixel out of the total activity along
the ith projection. If piis the measured projection, then the error is calcu-
lated as the difference (pi−qi), or as the ratio pi/qi. The weighting factors
are then applied to distribute this error (pi−qior pi/qi) into all pixels (N)
along the ith projection according to

(12.5)

where DCjis the error introduced into the jth pixel from all pixels in the ith
projection. Note that in error calculation, only pixels belonging to the same
projection have been considered. However, in reality, all image pixels have
a finite probability of contributing counts to any pixel in any projection and
the computation of errors becomes very time consuming.
There are three ways of calculating and applying error corrections. In a
point-by-point correction technique, the errors due to all pixels from all
projections passing through a particular pixel are calculated and used to
correct that pixel before proceeding to the next pixel. In a projection-by-
projection correction technique, the error is computed for each projection
and the image is updated before proceeding to the next projection. In the
simultaneous iteration technique, errors for all projections are computed
which are then applied simultaneously to update the image.
The two iterative algorithms widely used in image reconstruction are the
maximum-likelihood expectation maximization(MLEM) algorithm and the
ordered subset expectation maximization (OSEM) algorithm. The main
feature of the MLEM algorithm is to update the image during each itera-
tion using Eqs. (12.4) and (12.5). This method requires many iterations to
achieve a satisfactory agreement between the estimated and measured
images, demanding a lengthy computation time. To circumvent this
problem, the OSEM algorithm has been introduced, which is a modifica-
tion of MLEM in that projections are grouped into a number of subsets
separated by some fixed angle. For each subset, MLEM is applied and the
expected projection values are computed from the estimation of the pixel
values in all projections in the subset and compared with the measured
image. The variance is entered into the next subset, MLEM is applied, and
the image is updated. This is repeated for all subsets. After all subsets are
exhausted, a single iteration is considered complete. Such iteration is
repeated until an expected agreement is achieved between the estimated
and measured images. It has been shown that if there are nsubsets, and
once all subsets are used in a single iteration of the OSEM, an estimate is
produced which is similar to that obtained by niterations of the MLEM
using all projections. It is this property of the OSEM that accelerates the

ΔΔC

ap q

a

C

apq

a

j

ij i i

ij

j

N j

ij i i

ij

j

= N

()−

=

()

==

∑∑

11

or

168 12. Single Photon Emission Computed Tomography