Exact Localization and Superresolution with Noisy Data and Random Illumination

This paper studies the problem of exact localization of sparse (point or extended) objects with noisy data. The crux of the proposed approach consists of random illumination. Several recovery methods are analyzed: the Lasso, BPDN and the One-Step Thresholding (OST). For independent random probes, it is shown that both recovery methods can localize exactly $s=\cO(m)$, up to a logarithmic factor, objects where $m$ is the number of data. Moreover, when the number of random probes is large the Lasso with random illumination has a performance guarantee for superresolution, beating the Rayleigh resolution limit. Numerical evidence confirms the predictions and indicates that the performance of the Lasso is superior to that of the OST for the proposed set-up with random illumination.

💡 Research Summary

The paper addresses the fundamental problem of locating multiple sparse objects (point-like or extended) from noisy measurements, aiming to achieve exact support recovery and super‑resolution beyond the classical Rayleigh limit. The authors propose a novel measurement scheme based on random illumination (random phase modulation) and analyze three reconstruction algorithms: One‑Step Thresholding (OST), the Lasso, and Basis Pursuit Denoising (BPDN).

Problem setting. The data model is (Y = \Phi X + E) with (|E|_2 \le \epsilon). The sensing matrix (\Phi) has unit‑norm columns, and the goal is to recover the sparse vector (X) (support and amplitudes). Traditional greedy methods such as Orthogonal Matching Pursuit guarantee exact support recovery only for sparsity (s = O(\sqrt{m})) under coherence constraints, which is far below the desired (s = O(m)).

Algorithmic analysis.

• OST computes the matched filter (Z = \Phi^{}Y) and selects indices where (|Z_i| > \tau^{}). Under worst‑case coherence (\mu(\Phi) \le c_1\sqrt{m}) and average coherence (\nu(\Phi) \le \frac12 \mu(\Phi)\sqrt{m}), Proposition 2 shows that OST recovers exactly (s = O(m/\log N)) non‑zero entries with probability at least (1-9/N), provided the minimal signal amplitude exceeds a noise‑dependent threshold.
• Lasso solves (\min_Z \frac12|Y-\Phi Z|_2^2 + \gamma\sigma|Z|_1). Proposition 3 states that if (\mu(\Phi) \le a_0\log N) and (s \le c_0|\Phi|_2^2\log N), then with (\gamma = 2\sqrt{2\log N}) the Lasso recovers both support and sign exactly with probability (1-O(N^{-1})). This result permits sparsity up to (s = O(m)).
• BPDN is discussed via the Restricted Isometry Property (RIP). Proposition 4 gives an error bound (|\hat X - X|2 \le C_1 s^{-1/2}|X-X{(s)}|1 + C_2\epsilon) when (\delta{2s}<\sqrt{2}-1). While BPDN does not guarantee exact support recovery, it provides robust error control under more general conditions.

Random illumination. The key innovation is to illuminate the object plane with (p) independent random phase masks. Each probe contributes a factor (e^{i\theta_{kj}}) with (\theta_{kj}\sim\mathcal{U}