A major advantage with algebraic reconstruction methods over the
filtered back-projection method is that algebraic methods can be
adapted easily to account for non-ideal responses of the imaging
system. As an example, consider the problem of attenuation in emission
tomography, assuming that the attenuation profile of the object being
imaged is known. The reconstruction problem can be formulated in the
same fashion as described in the previous section, but the pixel
weights assigned by the projection operator 
will now depend on
the distance between the pixel and the detector, and the assumed
attenuation profile. Unlike Chang's method, which involves averaging
correction factors, this method allows an exact attenuation
correction. The matrix W used for the reconstructions in this
project has been corrected for attenuation using the same assumptions
(uniform attenuation within the phantom) as in the case of the
filtered back projection method, allowing a direct comparison of the
results. If a real attenuation profile were known, however, it could
be corrected for in the same way. Interestingly, some of the most
advanced gamma cameras available today provide this information by
performing simultaneous SPECT and transmission scans using multiple
energy windows.
The formation of the matrix W and the calculation of the appropriate weights was found to be one of the more time consuming parts of this algorithm, since it required loops nested in several levels. This was therefore implemented in C in order to reduce computation time.
The ability of the algebraic method to correct for imperfections in the imaging system are not limited to attenuation. If the response of the imaging system deviates from the ideal Radon transform for other reasons, such as the geometric arrangement of the detectors, the general matrix formulation still applies. This is in contrast to the filtered back-projection method which is based on the inverse Radon transform.