Factorization of the translation kernel for fast rigid image alignment

Abstract

An important component of many image alignment methods is the calculation of inner products (correlations) between an image of n×n pixels and another image translated by some shift and rotated by some angle. For robust alignment of an image pair, the number of considered shifts and angles is typically high, thus the inner product calculation becomes a bottleneck. Existing methods, based on fast Fourier transforms (FFTs), compute all such inner products with computational complexity O(n 3 log n) per image pair, which is reduced to O(N n 2 ) if only N distinct shifts are needed. We propose to use a factorization of the translation kernel (FTK), an optimal interpolation method which represents images in a Fourier-Bessel basis and uses a rank-H approximation of the translation kernel via an operator singular value decomposition (SVD). Its complexity is O(Hn(n + N )) per image pair. We prove that H = O((W + log(1/ )) 2 ), where 2W is the magnitude of the maximum desired shift in pixels and is the desired accuracy. For fixed W this leads to an acceleration when N is large, such as when sub-pixel shift grids are considered. Finally, we present numerical results in an electron cryomicroscopy application showing speedup factors of 3-10 with respect to the state of the art.

Unsupervised particle sorting for high-resolution single-particle cryo-EM

“…One drawback of this method, however, is the elevated computational cost required to evaluate the marginalized likelihood over all possible rotations and translations for each experimental image. Under certain conditions, however, the computational load may be alleviated by employing multiscale methods or advanced approximation techniques [62].…”

Radial-recombination for rigid rotational alignment of images and volumes

“…Many of the conventions we establish for the former will carry over to the latter. When possible, we will use the same notation as in [25].…”

“…As a consequence of Plancherel's theorem, any inner-product between A and B can be calculated equally well in either real-or frequency-space. Because of recent image-alignment tools developed in the context of cryo-EM molecular-reconstruction [27,25], we will typically refer to images in frequency-space (i.e., Â, rather than A). Nevertheless, all of the concepts we develop can be applied just as well in real-space [15,30,8,27,25].…”

“…Because of recent image-alignment tools developed in the context of cryo-EM molecular-reconstruction [27,25], we will typically refer to images in frequency-space (i.e., Â, rather than A). Nevertheless, all of the concepts we develop can be applied just as well in real-space [15,30,8,27,25].…”

“…Thus, while the set of transformations to be considered still includes all rotations, the translations can be limited to small translations with a bounded magnitude. In this setting the translational degreesof-freedom can be treated efficiently [25], and the problem reduces to a sub-problem where only rigidrotations need be considered.…”