A method is presented that directly calculates the mean number of scat
tered coincidences in data acquired with fully 3D positron emission to
mography (PET). This method uses a transmission scan, an emission scan
, the physics of Compton scatter, and a mathematical model of the scan
ner in a forward calculation of the number of events for which one pho
ton has undergone a single Compton interaction. The distribution of ev
ents for which multiple Compton interactions have occurred is modelled
as a linear transformation of the single-scatter distribution. Comput
ational efficiency is achieved by sampling at rates no higher than tho
se required by the scatter distribution and by implementing the algori
thm using look-up tables. Evaluation studies in phantoms with large sc
atter fractions show that the method yields images with quantitative a
ccuracy equivalent to that of slice-collimated PET in clinically usefu
l times.