POCS-based framework of signal reconstruction from generalized non-uniform samples

We formalize the use of projections onto convex sets (POCS) for the reconstruction of signals from non-uniform samples in their highest generality. This covers signals in any Hilbert space $\mathscr H$, including multi-dimensional and multi-channel signals, and samples that are most generally inner products of the signals with given kernel functions in $\mathscr H$. An attractive feature of the POCS method is the unconditional convergence of its iterates to an estimate that is consistent with the samples of the input, even when these samples are of very heterogeneous nature on top of their non-uniformity, and/or under insufficient sampling. Moreover, the error of the iterates is systematically monotonically decreasing, and their limit retrieves the input signal whenever the samples are uniquely characteristic of this signal. In the second part of the paper, we focus on the case where the sampling kernel functions are orthogonal in $\mathscr H$, while the input may be confined in a smaller closed space $\mathscr A$ (of bandlimitation for example). This covers the increasingly popular application of time encoding by integration, including multi-channel encoding. We push the analysis of the POCS method in this case by giving a special parallelized version of it, showing its connection with the pseudo-inversion of the linear operator defined by the samples, and giving a multiplierless discrete-time implementation of it that paradoxically accelerates the convergence of the iteration.

Bandlimited signal reconstruction from leaky integrate-and-fire encoding using POCS

Leaky integrate-and-fire (LIF) encoding is a model of neuron transfer function in biology that has recently attracted the attention of the signal processing and neuromorphic computing communities as a technique of event-based sampling for data acquisition. While LIF enables the implementation of analog-circuit signal samplers of lower complexity and higher accuracy simultaneously, the core difficulty of this technique is the retrieval of an input from its LIF-encoded output. This theoretically requires to perform the pseudo-inversion of a linear but time-varying operator of virtually infinite size. In the context of bandlimited inputs to allow finite-rate sampling, we show two fundamental contributions of the method of projection onto convex sets (POCS) to this problem: (i) single iterations of the POCS method can be used to deterministically improve input estimates from any other reconstruction method; (ii) the iteration limit of the POCS method is the pseudo-inverse of the above mentioned operator in all conditions, whether reconstruction is unique or not and whether the encoding is corrupted by noise or not. The algorithms available until now converge only under particular situations of unique of reconstruction.