Radial-recombination for rigid rotational alignment of images and volumes

Abstract

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 innerproduct between one image of N × N pixels and another image that has been translated by some displacement and rotated by some angle γ. In many situations the number of rotations γ considered is large (e.g., O(N )), while the number of translations considered is much smaller (e.g., O(1)). In these scenarios a naive algorithm requires 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 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 O(N (log(N ) + H)) per image-pair, where H is the rank of the SVD used in the compression above. A very similar strategy can be used to accelerate volume-alignment, using a spherical-harmonic based compression, which we'll refer to as a 'degree-SVD'. 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.