  Principal component analysis (PCA) plays an important role in the analysis of
cryo-EM images for various tasks such as classification, denoising,
compression, and ab-initio modeling. We introduce a fast method for estimating
a compressed representation of the 2-D covariance matrix of noisy cryo-electron
microscopy projection images that enables fast PCA computation. Our method is
based on a new algorithm for expanding images in the Fourier-Bessel basis (the
harmonics on the disk), which provides a convenient way to handle the effect of
the contrast transfer functions. For $N$ images of size $L\times L$, our method
has time complexity $O(N L^3 + L^4)$ and space complexity $O(NL^2 + L^3)$. In
contrast to previous work, these complexities are independent of the number of
different contrast transfer functions of the images. We demonstrate our
approach on synthetic and experimental data and show acceleration by factors of
up to two orders of magnitude.
