Motivated by modern data applications such as cryo-electron microscopy, the goal of classic multi-reference alignment (MRA) is to recover an unknown signal $f: \mathbb{R} \to \mathbb{R}$ from many observations that have been randomly translated and corrupted by additive noise. We consider a generalization of classic MRA where signals are also corrupted by a random scale change, i.e. dilation. We propose a novel data-driven unbiasing procedure which can recover an unbiased estimator of the bispectrum of the unknown signal, given knowledge of the dilation distribution. Lastly, we invert the recovered bispectrum to achieve full signal recovery, and validate our methodology on a set of synthetic signals.

翻译：受冷冻电子显微镜等现代数据应用的启发，经典联合对齐（MRA）的目标是从大量经过随机平移并被加性噪声破坏的观测中恢复未知信号$f: \mathbb{R} \to \mathbb{R}$。我们考虑了经典MRA的推广形式，其中信号还受到随机尺度变化（即膨胀）的破坏。我们提出了一种新颖的数据驱动无偏估计流程，该流程在已知膨胀分布的情况下，能够恢复未知信号双谱的无偏估计量。最后，我们通过反演恢复的双谱实现完整信号恢复，并在合成信号集上验证了该方法。