The most used algorithm in emission tomography is the ML-EM (Maximum Likelihood Expectation Maximization) (Shepp and Vardi, 1982) and it's accelerated version OS-EM (Hudson and Larkin, 1994).
I think in the literature, and depending on the scanner (mainly at preclinical level), the most used algorithm are OSEM algorithms. I put algorithms plural because of the different variants that exists (2D, 3D, +/- MAP or Fast MAP, etc..). I am not too sure that all algorithms provided by scanner manufacturers although called OSEM are all identical as it seems quite difficult sometimes to really know what is behind that name.
OSEM is the standard in clinical PET scanners and pre-clinical PET systems using manufacturer delivered reconstruction software or locally written software. There is quite a bit of free code available to get you started.
First check this chapter on Free Software for PET - http://www.intechopen.com/books/positron-emission-tomography-current-clinical-and-research-aspects/free-software-for-pet-imaging
http://www.intechopen.com/books/positron-emission-tomography-current-clinical-and-research-aspects/free-software-for-pet-imaging
Second - two handy packages come to mind as a place to start - NiftyRec and STIR.
Niftyrec is a tomography recon package from UCL . Its available here. http://niftyrec.scienceontheweb.net/wordpress/
STIR - Software for Tomographic Image Reconstruction - is here - http://stir.sourceforge.net/hompage.shtml
Both packages have extensive documentation and information to allow you to get started quickly. Make sure to check the links page on the STIR site. It has a variety of pointers to other information to get you started.
We are currently testing various reconstructions software in Manchester, and so far we have found out that the algorithms provided by manufacturers are only tested (at preclinical level) on static phantom, whereas we use dynamic histogramming with short time frames... and here the algorithms do not always provide the answer they should... We hope to publish this soon, but this is an incredible tedious task.... So the take home message is: check the reconstructions methods you want to use on phantoms for which you know the true values in setting that reproduce your "real" scanning conditions to avoid disappointment!
I reviewed reconstruction algorithms in the following publication:
Conference Paper A REVIEW OF RECENT RANGE IMAGE RECONSTRUCTION ALGORITHMS
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