# Factorization of the translation kernel for fast rigid image alignment

## TL;DR

This paper introduces a novel factorization method for the translation kernel that significantly accelerates the computation of inner products in image alignment, especially for large shift sets, demonstrated by a 3-10x speedup in cryo-EM applications.

## Contribution

The paper proposes a Fourier--Bessel basis-based factorization of the translation kernel, reducing computational complexity for image alignment tasks compared to FFT-based methods.

## Key findings

- Achieves 3-10x speedup in electron cryomicroscopy image alignment.
- Provides a theoretical bound on the approximation rank H for the translation kernel.
- Demonstrates efficiency gains when considering large sets of shifts in image registration.

## Abstract

An important component of many image alignment methods is the calculation of inner products (correlations) between an image of $n\times n$ 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 $\mathcal{O}(n^3 \log n)$ per image pair, which is reduced to $\mathcal{O}(N n^2)$ 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 $\mathcal{O}(Hn(n + N))$ per image pair. We prove that $H = \mathcal{O}((W + \log(1/\epsilon))^2)$, where $2W$ is the magnitude of the maximum desired shift in pixels and $\epsilon$ 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.