  In cryo-electron microscopy, the 3D electric potentials of an ensemble of
molecules are projected along arbitrary viewing directions to yield noisy 2D
images. The volume maps representing these potentials typically exhibit a great
deal of structural variability, which is described by their 3D covariance
matrix. Typically, this covariance matrix is approximately low-rank and can be
used to cluster the volumes or estimate the intrinsic geometry of the
conformation space. We formulate the estimation of this covariance matrix as a
linear inverse problem, yielding a consistent least-squares estimator. For $n$
images of size $N$-by-$N$ pixels, we propose an algorithm for calculating this
covariance estimator with computational complexity
$\mathcal{O}(nN^4+\sqrt{\kappa}N^6 \log N)$, where the condition number
$\kappa$ is empirically in the range $10$--$200$. Its efficiency relies on the
observation that the normal equations are equivalent to a deconvolution problem
in 6D. This is then solved by the conjugate gradient method with an appropriate
circulant preconditioner. The result is the first computationally efficient
algorithm for consistent estimation of 3D covariance from noisy projections. It
also compares favorably in runtime with respect to previously proposed
non-consistent estimators. Motivated by the recent success of eigenvalue
shrinkage procedures for high-dimensional covariance matrices, we introduce a
shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We
evaluate our methods on simulated datasets and achieve classification results
comparable to state-of-the-art methods in shorter running time. We also present
results on clustering volumes in an experimental dataset, illustrating the
power of the proposed algorithm for practical determination of structural
variability.
