  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.
