Often a set of images is obtained from a heterogeneous mixture of particles of two or more different kinds or different conformations. It is therefore desirable to reconstruct not just a single averaged volume, but the entire set of 3D molecular conformations. We have developed a method for estimating the covariance matrix of the distribution of the 3D volumes directly from the 2D projection images.  This functionality is currently available to download from an external Github repository, and it will soon be integrated into the ASPIRE package.  We recently developed two new approaches for analyzing continuous heterogeneity: a manifold learning approach based on Laplacian spectral volumes, and a hyper-molecules representation for joint estimation of orientations and variability. Software for these approaches is not yet publicly available.

Further reading:

A. Moscovich, A. Halevi, J. Andén, A. Singer,  Cryo-EM reconstruction of continuous heterogeneity by Laplacian spectral volumes, submitted.

R. R. Lederman, J. Andén, A. Singer,  Hyper-Molecules: on the Representation and Recovery of Dynamical Structures, with Application to Flexible Macro-Molecular Structures in Cryo-EM, submitted.

J. Andén, A. Singer, Structural Variability from Noisy Tomographic Projections, SIAM Journal on Imaging Sciences, 11(2), pp. 1441-1492, 2018. 

R. R. Lederman, A. Singer, A Representation Theory Perspective on Simultaneous Alignment and Classification, arXiv preprint.

J. Andén, E. Katsevich, A. Singer, Covariance estimation using conjugate gradient for 3D classification in Cryo-EM, in IEEE 12th International Symposium on Biomedical Imaging (ISBI 2015), pp. 200-204, 16-19 April 2015.

E. Katsevich, A. Katsevich, A. Singer, Covariance Matrix Estimation for the Cryo-EM Heterogeneity Problem, SIAM Journal on Imaging Sciences, 8 (1), pp. 126-185 (2015).