Bandlimited signal reconstruction from orthogonally-projected data

We show that a broad class of signal acquisition schemes can be interpreted as recording data from a signal $x$ in a space $\cal U$ (typically, though not exclusively, a space of bandlimited functions) via an orthogonal projection $w = P_{\cal V} x$ onto another space $\cal V$. A basic reconstruction method in this case consists in alternating projections between the input space $\cal U$ and the affine space $\cal W$ of signals $u$ satisfying $P_{\cal V} u = w$ (POCS method). Although this method is classically known to be slow, our work reveals new insights and contributions: (i) it applies to new complex encoders emerging from event-based sampling, for which no faster reconstruction method is currently available; (ii) beyond perfect reconstruction, it converges robustly under insufficient (e.g., sub-Nyquist) or inconsistent data (due to noise or errors); (iii) the limit of convergence achieves optimal least-squares approximations under such conditions; (iv) semi-convergence inherently results in regularized reconstructions under ill-posed data acquisition conditions; (v) when $w$ is produced by discrete sampling, the iterative method can be rigorously discretized for DSP implementation -- even with non-separable input spaces $\cal U$. While moving beyond the traditional focus on perfect reconstruction in harmonic analysis, our analysis preserves the deterministic framework of infinite-dimensional Hilbert spaces, consistent with Shannon's sampling theory. For illustration, we apply our proposed theory to two contrasting sampling situations: multi-channel time encoding and nonuniform point sampling in a Sobolev space.

Pseudo-inverse reconstruction of bandlimited signals from nonuniform generalized samples with orthogonal kernels

Contrary to the traditional pursuit of research on nonuniform sampling of bandlimited signals, the objective of the present paper is not to find sampling conditions that permit perfect reconstruction, but to perform the best possible signal recovery from any given set of nonuniform samples, whether it is finite as in practice, or infinite to achieve the possibility of unique reconstruction in $L^2({\bf R})$. This leads us to consider the pseudo-inverse of the whole sampling map as a linear operator of Hilbert spaces. We propose in this paper an iterative algorithm that systematically performs this pseudo-inversion under the following conditions: (i) the input lies in some closed space $\cal A$ (such as a space of bandlimited functions); (ii) the samples are formed by inner product of the input with given kernel functions; (iii) these functions are orthogonal at least in a Hilbert space $\cal H$ that is larger than $\cal A$. This situation turns out to appear in certain time encoders that are part of the increasingly important area of event-based sampling. As a result of pseudo-inversion, we systematically achieve perfect reconstruction whenever the samples uniquely characterize the input, we obtain minimal-norm estimates when the sampling is insufficient, and the reconstruction errors are controlled in the case of noisy sampling. The algorithm consists in alternating two projections according to the general method of projections onto convex sets (POCS) and can be implemented by iterating time-varying discrete-time filtering. We finally show that our signal and sampling assumptions appear in a nontrivial manner in other existing problems of data acquisition. This includes multi-channel time encoding where $\cal H$ is of the type $L^2({\bf R})^M$, and traditional point sampling with the adoption of a Sobolev space $\cal H$.

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.