3-D reconstructions of nanometer-scale helical objects from cryo electron microscopy images

Statistical algorithms are described for 3-D signal reconstruction problems where the object has a helical symmetry with unknown symmetry parameters and the data is cryo electron micrographs of multiple identical instances of the object. Two object models are described: a Fourier-Bessel series model for a helical object and a spherical harmonics series model for a motif which is then replicated in an array to form a helical object where, in both cases, the unknown parameters are the coefficients in the series. The measured images are essentially noisy 2-D projections of the 3-D electron scattering intensity where the projection orientation is not known. A maximum likelihood estimator computed by an expectation-maximization algorithm is used in both cases to estimate the unknown parameters.