  This revisit gives a survey on the analytical methods for the inverse
exponential Radon transform which has been investigated in the past three
decades from both mathematical interests and medical applications such as
nuclear medicine emission imaging. The derivation of the classical inversion
formula is through the recent argument developed for the inverse attenuated
Radon transform. That derivation allows the exponential parameter to be a
complex constant, which is useful to other applications such as magnetic
resonance imaging and tensor field imaging. The survey also includes the new
technique of using the finite Hilbert transform to handle the exact
reconstruction from 180 degree data. Special treatment has been paid on two
practically important subjects. One is the exact reconstruction from partial
measurements such as half-scan and truncated-scan data, and the other is the
reconstruction from diverging-beam data. The noise propagation in the
reconstruction is touched upon with more heuristic discussions than
mathematical inference. The numerical realizations of several classical
reconstruction algorithms are included. In the conclusion, several topics are
discussed for more investigations in the future.
