Bandlimited signal reconstruction from orthogonally-projected data

Abstract

We show that a broad class of signal acquisition schemes can be interpreted as recording data from a signal $x$ in a space $\mathcal{U}$ (typically, though not exclusively, a space of bandlimited functions) via an orthogonal projection $w=P_{\mathcal{V}}x$ onto another space $\mathcal{V}$. A basic reconstruction method in this case consists in alternating projections between the input space $\mathcal{U}$ and the affine space $\mathcal{W}$ of signals $u$ satisfying $P_{\mathcal{V}}u=w$ (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 $\mathcal{U}$. 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.

13 pages, 3 figures