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 $\cal H$.