The intrinsic detector efficiencies are again usually computed from an acquisition of a central uniform right cylinder source, although planar or rotating line sources can also be used. Variance reduction may be effected using the fan-sum algorithm, which is essentially a simplified version of that used in randoms variance reduction. In the fan-sum algorithm, the fans of LORs emanating from each detector and defining a group A of opposing detectors are summed (see Fig. 5.11). It is assumed that the activity distribution intersected by each fan is the same, and that the effect of all normalisation components apart from detector efficiency is also the same for each fan. The total counts in each fan Cui then obeys the following relation:
If A contains a sufficiently large number of detectors, it can be assumed that the expression ^ ^ £v is also a
L veA jeA
constant (the fan-sum approximation, attributable to 17). The £ui are then given by the following expression:
where Cui is the mean value of all the fan-sums for detector ring u. Note that the efficiencies are not determined using the mean value of the Cui computed over all detector rings as the numerator in equation (16). This avoids potential bias arising from the fact that the mean angle of incidence of the LORs at the detector face varies from the axial centre to the front or back of the tomograph.
If the Cui are calculated by summing only over LORs lying within detector ring u, the method is known as the 2D fan-sum algorithm. This method is quite widely implemented because of its simplicity, and because it can be used for both 2D and 3D normalization. However, in the 3D case it is both less accurate and less precise than utilizing all possible LORs . The accuracy of the fan-sum approximation also depends crucially on utilizing an accurately centered source distribution [19-21]. Other algorithms for calculating the £ui also exist (see, for example, [17,18,20-22]).
The £ui calculated in this way incorporate the transaxial block profile factors bIf required, they can be extracted from the £ui very simply - they are just the mean values of the detector efficiencies calculated for each position in the block detector:
y where N is the number of detectors around the ring and D is the number of detectors across a detector block. In practice, the btur are obtained by averaging data across such a large and evenly sampled proportion of the field of view that they are effectively independent of the source distribution. As a result, changes in the transaxial block profile factors due to count-rate effects can be computed directly from the emission data, a process known as self-normalisation .
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