  A common task in single particle electron cryomicroscopy (cryo-EM) is the
rigid alignment of images and/or volumes. In the context of images, a rigid
alignment involves estimating the inner-product between one image of $N\times
N$ pixels and another image that has been translated by some displacement and
rotated by some angle $\gamma$. In many situations the number of rotations
$\gamma$ considered is large (e.g., $\mathcal{O}(N)$), while the number of
translations considered is much smaller (e.g., $\mathcal{O}(1)$). In these
scenarios a naive algorithm requires $\mathcal{O}(N^{3})$ operations to
calculate the array of inner-products for each image-pair. This computation can
be accelerated by using a fourier-bessel basis and the fast-fourier-transform
(FFT), requiring only $\mathcal{O}(N^2)$ operations per image-pair. We propose
a simple data-driven compression algorithm to further accelerate this
computation, which we refer to as the `radial-SVD'. Our approach involves
linearly-recombining the different rings of the original images (expressed in
polar-coordinates), taking advantage of the singular-value-decomposition (SVD)
to choose a low-rank combination which both compresses the images and optimizes
a certain measure of angular discriminability. When aligning multiple images to
multiple 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 bit
smaller than $N$ while still maintaining accuracy. We present numerical results
in a cryo-EM application demonstrating that the radial- and degree-SVD can help
save a factor of $5$--$10$ for both image- and volume-alignment.
