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A common task in single particle electron cryomicroscopy (cryo-EM) is therigid alignment of images and/or volumes. In the context of images, a rigidalignment involves estimating the inner-product between one image of $N\timesN$ pixels and another image that has been translated by some displacement androtated by some angle $\gamma$. In many situations the number of rotations$\gamma$ considered is large (e.g., $\mathcal{O}(N)$), while the number oftranslations considered is much smaller (e.g., $\mathcal{O}(1)$). In thesescenarios a naive algorithm requires $\mathcal{O}(N^{3})$ operations tocalculate the array of inner-products for each image-pair. This computation canbe accelerated by using a fourier-bessel basis and the fast-fourier-transform(FFT), requiring only $\mathcal{O}(N^2)$ operations per image-pair. We proposea simple data-driven compression algorithm to further accelerate thiscomputation, which we refer to as the `radial-SVD'. Our approach involveslinearly-recombining the different rings of the original images (expressed inpolar-coordinates), taking advantage of the singular-value-decomposition (SVD)to choose a low-rank combination which both compresses the images and optimizesa certain measure of angular discriminability. When aligning multiple images tomultiple targets, the complexity of our approach is $\mathcal{O}(N(\log(N)+H))$per image-pair, where $H$ is the rank of the SVD used in the compression above.The advantage gained by this approach depends on the ratio between $H$ and $N$;the smaller $H$ is the better. In many applications $H$ can be quite a bitsmaller than $N$ while still maintaining accuracy. We present numerical resultsin a cryo-EM application demonstrating that the radial- and degree-SVD can helpsave a factor of $5$--$10$ for both image- and volume-alignment.

Radial-recombination for rigid rotational alignment of images and volumes