  Cryo-electron microscopy provides 2-D projection images of the 3-D electron
scattering intensity of many instances of the particle under study (e.g., a
virus). Both symmetry (rotational point groups) and heterogeneity are important
aspects of biological particles and both aspects can be combined by describing
the electron scattering intensity of the particle as a stochastic process with
a symmetric probability law and therefore symmetric moments. A maximum
likelihood estimator implemented by an expectation-maximization algorithm is
described which estimates the unknown statistics of the electron scattering
intensity stochastic process from images of instances of the particle. The
algorithm is demonstrated on the bacteriophage HK97 and the virus N$\omega$V.
The results are contrasted with existing algorithms which assume that each
instance of the particle has the symmetry rather than the less restrictive
assumption that the probability law has the symmetry.
