Radial-recombination for rigid rotational alignment of images and volumes

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.