The multi-reference alignment (MRA) problem entails estimating an image from multiple noisy and rotated copies of itself. If the noise level is low, one can reconstruct the image by estimating the missing rotations, aligning the images, and averaging out the noise. While accurate rotation estimation is impossible if the noise level is high, the rotations can still be approximated, and thus can provide indispensable information. In particular, learning the approximation error can be harnessed for efficient image estimation. In this paper, we propose a new computational framework, called Synch-EM, that consists of angular synchronization followed by expectation-maximization (EM). The synchronization step results in a concentrated distribution of rotations; this distribution is learned and then incorporated into the EM as a Bayesian prior. The learned distribution also dramatically reduces the search space, and thus the computational load of the EM iterations. We show by extensive numerical experiments that the proposed framework can significantly accelerate EM for MRA in high noise levels, occasionally by a few orders of magnitude, without degrading the reconstruction quality.

中文翻译：

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多参考对齐的加速期望最大化算法


多参考对齐 (MRA) 问题需要根据图像自身的多个噪声和旋转副本来估计图像。如果噪声水平较低，则可以通过估计丢失的旋转、对齐图像并对噪声进行平均来重建图像。虽然如果噪声水平较高则不可能进行精确的旋转估计，但仍然可以近似旋转，从而可以提供不可或缺的信息。特别是，可以利用学习近似误差来进行有效的图像估计。在本文中，我们提出了一种新的计算框架，称为 Synch-EM，它由角度同步和期望最大化（EM）组成。同步步骤导致旋转的集中分布；学习该分布，然后将其作为贝叶斯先验合并到 EM 中。学习到的分布还极大地减少了搜索空间，从而减少了 EM 迭代的计算负载。我们通过大量的数值实验表明，所提出的框架可以在高噪声水平下显着加速 MRA 的 EM，有时会加速几个数量级，而不会降低重建质量。