Mismatches between samples and their respective channel or target commonly arise in several real-world applications. For instance, whole-brain calcium imaging of freely moving organisms, multiple-target tracking or multi-person contactless vital sign monitoring may be severely affected by mismatched sample-channel assignments. To systematically address this fundamental problem, we pose it as a signal reconstruction problem where we have lost correspondences between the samples and their respective channels. Assuming that we have a sensing matrix for the underlying signals, we show that the problem is equivalent to a structured unlabeled sensing problem, and establish sufficient conditions for unique recovery. To the best of our knowledge, a sampling result for the reconstruction of shuffled multi-channel signals has not been considered in the literature and existing methods for unlabeled sensing cannot be directly applied. We extend our results to the case where the signals admit a sparse representation in an overcomplete dictionary (i.e., the sensing matrix is not precisely known), and derive sufficient conditions for the reconstruction of shuffled sparse signals. We propose a robust reconstruction method that combines sparse signal recovery with robust linear regression for the two-channel case. The performance and robustness of the proposed approach is illustrated in an application related to whole-brain calcium imaging. The proposed methodology can be generalized to sparse signal representations other than the ones considered in this work to be applied in a variety of real-world problems with imprecise measurement or channel assignment.

翻译：在多个实际应用中，样本与其各自通道或目标之间的失配现象普遍存在。例如，自由运动生物的全脑钙成像、多目标追踪或多人生理体征非接触监测可能受到样本-通道分配失配的严重影响。为系统解决这一基本问题，我们将其建模为样本与对应通道之间对应关系缺失的信号重构问题。假设已知底层信号的感知矩阵，我们证明该问题等价于结构化未标记感知问题，并建立了唯一恢复的充分条件。据我们所知，关于打乱多通道信号重构的采样结果在文献中尚未被探讨，且现有的未标记感知方法无法直接应用。我们将研究结果推广至信号在过完备字典中具有稀疏表示（即感知矩阵不完全已知）的情形，推导了打乱稀疏信号重构的充分条件。针对双通道场景，我们提出了一种结合稀疏信号恢复与稳健线性回归的鲁棒重构方法。通过全脑钙成像应用实例，验证了所提方法的性能与鲁棒性。该方法论可推广至除本文所考虑之外的其他稀疏信号表示，用于处理多种测量或通道分配不精确的真实世界问题。