Unlabeled Sensing of Multi-Input FRI Signals

Shannon's sampling theorem for bandlimited signals, formulated in 1949, has become a cornerstone for modern digital communications and signal processing. The importance of sampling and reconstruction of analog signals has led to great advances in the field, such as sampling of nonbandlimited signals, compressed sensing, or unlabeled sensing, which go far beyond the uniform sampling of bandlimited signals. In this thesis, we address the problem of reconstructing multi-input signals from shuffled samples; a problem which has not yet been covered by the scientific literature. This sampling setup can occur in various real-world applications that include, but are not limited to, whole-brain calcium imaging and general fluorescence microscopy of moving cell cultures as well as radar entomology. We first study the case of two shuffled signals with finite rate of innovation, focusing on streams of decaying exponentials, and derive necessary conditions for unique reconstruction by drawing a link between the studied problem and previous work on signals with finite rate of innovation and linear regression without correspondences. Building on this derivation we propose a two-step estimation approach for unshuffling and reconstructing the signals of interest: Step 1 deals with the recovery of the signal parameters of the sum of the input signals. These parameters are fed to Step 2, which focuses on solving a structured unlabeled sensing problem. We devise a heuristic relying on concepts from robust estimation theory to iteratively estimate the signal parameters of the individual signals and reduce the number of shuffled samples. We provide a comprehensive evaluation of the proposed method in numerical experiments, where the influence of additive noise, the number of shuffled samples, the exponential decay, and the propagated errors from Step 1 to Step 2 are analyzed. In addition, we investigate a possible gain from reestimating the signal parameters from the estimated individual channels and present exemplary results on artificially shuffled calcium imaging traces. Furthermore, in a benchmark against an expectation maximization method for unlabeled sensing, we observe the proposed method's average signal reconstruction error to be multiple orders of magnitude smaller. The results illustrate the satisfactory performance of the proposed method, which to date is the first method developed to deal with the considered sampling setup.