Reconstruction of stochastic 3-D signals with symmetric statistics from 2-D projection images motivated by cryo-electron microscopy.

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.