  In cryo-electron microscopy (cryo-EM), a microscope generates a top view of a
sample of randomly-oriented copies of a molecule. The problem of single
particle reconstruction (SPR) from cryo-EM is to use the resulting set of noisy
2D projection images taken at unknown directions to reconstruct the 3D
structure of the molecule. In some situations, the molecule under examination
exhibits structural variability, which poses a fundamental challenge in SPR.
The heterogeneity problem is the task of mapping the space of conformational
states of a molecule. It has been previously suggested that the leading
eigenvectors of the covariance matrix of the 3D molecules can be used to solve
the heterogeneity problem. Estimating the covariance matrix is challenging,
since only projections of the molecules are observed, but not the molecules
themselves. In this paper, we formulate a general problem of covariance
estimation from noisy projections of samples. This problem has intimate
connections with matrix completion problems and high-dimensional principal
component analysis. We propose an estimator and prove its consistency. When
there are finitely many heterogeneity classes, the spectrum of the estimated
covariance matrix reveals the number of classes. The estimator can be found as
the solution to a certain linear system. In the cryo-EM case, the linear
operator to be inverted, which we term the projection covariance transform, is
an important object in covariance estimation for tomographic problems involving
structural variation. Inverting it involves applying a filter akin to the ramp
filter in tomography. We design a basis in which this linear operator is sparse
and thus can be tractably inverted despite its large size. We demonstrate via
numerical experiments on synthetic datasets the robustness of our algorithm to
high levels of noise.
