Radial-recombination for rigid rotational alignment of images and volumes
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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× 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. 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.
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