Bendory Tamir, Jaffe Ariel, Leeb William, Sharon Nir, Singer Amit
School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel.
Applied Mathematics Program, Yale University, New Haven, CT, USA.
Inf inference. 2022 Jun;11(2):533-555. doi: 10.1093/imaiai/iaab003. Epub 2021 Feb 18.
We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in is uniquely determined when the number of samples per observation is of the order of the square root of the signal's length ( ). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to 1/SNR. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled ( = ). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.
我们研究超分辨率多参考对齐问题，即从多个循环移位、下采样且有噪声的观测值中估计信号的问题。我们专注于低信噪比（SNR）情况，并表明当每个观测值的样本数量为信号长度的平方根量级（ ）时， 中的信号是唯一确定的。更通俗地说，分辨率可以提高到平方倍。如果观测值的数量与1/SNR成正比，该结果成立。相比之下，即使观测值没有下采样（ = ），观测值较少时也无法恢复信号。该分析结合了统计信号处理和不变量理论的工具。我们设计了一种期望最大化算法，并证明它能够在具有挑战性的SNR情况下对信号进行超分辨率处理。
