Unlocking Off-the-Grid Sparse Recovery with Unlimited Sensing: Simultaneous Super-Resolution in Time and Amplitude

TL;DR

The paper addresses the challenge of recovering off-the-grid sparse spikes from folded, low-bit measurements by leveraging the Unlimited Sensing Framework (USF) to achieve simultaneous amplitude and temporal super-resolution. It advances theory for non-bandlimited kernels, introduces a robust SRes algorithm (SRes-IterSiS) that decomposes the problem into residue and spike-estimation subproblems, and validates the approach with ToF imaging experiments showing significant dynamic-range enhancements and centimeter-level depth accuracy using as few as 3 bits. The combination of theory, robust algorithm design, and hardware validation demonstrates that modulo encoding can unlock digital super-resolution beyond conventional limits, with practical impact for HDR sensing and low-power imaging. The work foregrounds USF as a viable foundation for joint amplitude-time super-resolution in resource-constrained sensing systems.

Abstract

The recovery of Dirac impulses, or spikes, from filtered measurements is a classical problem in signal processing. As the spikes lie in the continuous domain while measurements are discrete, this task is known as super-resolution or off-the-grid sparse recovery. Despite significant theoretical and algorithmic advances over the past decade, these developments often overlook critical challenges at the analog-digital interface. In particular, when spikes exhibit strong-weak amplitude disparity, conventional digital acquisition may result in clipping of strong components or loss of weak ones beneath the quantization noise floor. This motivates a broader perspective: super-resolution must simultaneously resolve both amplitude and temporal structure. Under a fixed bit budget, such information loss is unavoidable. In contrast, the emerging theory and practice of the Unlimited Sensing Framework (USF) demonstrate that these fundamental limitations can be overcome. Building on this foundation, we demonstrate that modulo encoding within USF enables digital super-resolution by enhancing measurement precision, thereby unlocking temporal super-resolution beyond conventional limits. We develop new theoretical results that extend to non-bandlimited kernels commonly encountered in practice and introduce a robust algorithm for off-the-grid sparse recovery. To demonstrate practical impact, we instantiate our framework in the context of time-of-flight imaging. Both numerical simulations and hardware experiments validate the effectiveness of our approach under low-bit quantization, enabling super-resolution in amplitude and time.

Figures (9)

• Figure 1: Performance evaluation across varying dynamic ranges (DR) and quantization bits. The measurements follow the model $g[n] = \Gamma[1]\psi(nT - \tau[1]) + \Gamma[2]\psi(nT - \tau[2])$ (see \ref{['eq:g']}), involving a weak-strong mixture with an amplitude ratio of $\left| \Gamma \left[ 1 \right]/\Gamma \left[ 2 \right]\right| = 10$. The detailed experimental setup is described in Section \ref{['subsec:HDR']}. We compare the recovery mean-squared error (MSE) between a conventional ADC and USF across quantization levels ($\left[ 3,15 \right]$) and input DR values ($\| {g} \|_{{\mathsf{L}}_{{\infty}}} = \{10, 20, 30\}\lambda$). In all scenarios, USF consistently achieves a performance gain of at least $30$ dB and demonstrates robustness to input DR variations.
• Figure 2: Block diagram for modulo ToF image formation process. The goal is to estimate $s_{\mathbf{r}}\left( t \right)$ from $\{y_{\mathbf{r}} \left[ n \right]\}_{n\in\mathbb{I}_{N} }$.
• Figure 3: Experimental setup for SRes ToF Imaging. The $3$D scene comprises of a mannequin head positioned between a diffusive semi-translucent surface and a wall in the backdrop. The diffusive sheet is moving closer to the mannequin head, leading to the challenges of separating two close objects.
• Figure 4: Clipping-free HDR recovery ($\| {g} \|_{{\mathsf{L}}_{{\infty}}} = 20\lambda$). On the acquisition front, the DR of the conventional ADC is the same as the 0.9$\mathscr{M}_\lambda$--ADC, where the resulting data clipping causes inaccurate time-delay estimation. On the algorithm front, the FP-SR approach suffers from spectral leakage and dense folds, leading to large deviation in $3$D scene recovery. In contrast, the proposed method offers accurate signal recovery, where the estimated time-delays are visually indistinguishable from the ground-truth.
• Figure 5: SRes ToF imaging. The $3$D scene consists of a mannequin head placed between a diffusive translucent surface and a wall (backdrop), as shown in Fig. \ref{['fig:Scene']}. By moving the diffusive surface, the inter-object separation is uniformly reducing from (a) $1.6$ m, (b) $1.3$ m, (c) $1.0$ m to (d) $0.7$ m, respectively. Using $3$-bits quantization, our method achieves $\| {g} \|_{{\mathsf{L}}_{{\infty}}}=10\lambda$ and super-resolves the inter-object separation with estimation error down to $0.9$ cm.

Theorems & Definitions (5)

• Theorem 1: Kolmogorov Kolmogorov:1949:J
• Lemma 1
• proof
• Theorem 2
• proof