It is well known that a band-limited signal can be reconstructed from its uniformly spaced samples if the sampling rate is sufficiently high. More recently, it has been proved that one can reconstruct a 1D band-limited signal even if the exact sample locations are unknown, but given just the distribution of the sample locations and their ordering in 1D. In this work, we extend the analytical bounds on the reconstruction error in such scenarios for quasi-bandlimited signals. We also prove that the method for such a reconstruction is resilient to a certain proportion of errors in the specification of the sample location ordering. We then express the problem of tomographic reconstruction of 2D images from 1D Radon projections under unknown angles with known angle distribution, as a special case for reconstruction of quasi-bandlimited signals from samples at unknown locations with known distribution. Building upon our theoretical background, we present asymptotic bounds for 2D quasi-bandlimited image reconstruction from 1D Radon projections in the unknown angles setting, which commonly occurs in cryo-electron microscopy (cryo-EM). To the best of our knowledge, this is the first piece of work to perform such an analysis for 2D cryo-EM, even though the associated reconstruction algorithms have been known for a long time.

翻译：众所周知，若采样率足够高，带限信号可由其均匀采样点进行重建。近年来已有研究证明，即使一维带限信号的精确采样位置未知，仅凭采样点分布及其一维排序信息即可实现重建。本文将此情形下准带限信号的重建误差分析界限进行了拓展，并证明了该类重建方法对采样点排序信息中一定比例的误差具有鲁棒性。进一步地，我们将一维拉东投影在未知角度但已知角度分布条件下的二维图像断层重建问题，表述为已知分布未知采样位置条件下准带限信号重建的特例。基于理论框架，我们提出了在未知角度设定下（该设定常见于冷冻电镜领域）利用一维拉东投影进行二维准带限图像重建的渐进界。据我们所知，尽管相关重建算法早已为人所知，但本文是首次针对二维冷冻电镜开展此类分析的工作。