Stable and robust sampling strategies for compressive imaging

In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence – the so-called local coherence – measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by $\ell_1$-minimization and by total variation minimization. The local coherence framework developed in this paper should be of independent interest in sparse recovery problems more generally, as it implies that for optimal sparse recovery results, it suffices to have bounded \emph{average} coherence from sensing basis to sparsity basis – as opposed to bounded maximal coherence – as long as the sampling strategy is adapted accordingly.

💡 Research Summary

This paper addresses a fundamental gap in compressive imaging theory: the lack of incoherence between the Fourier measurement basis and the wavelet sparsity basis, particularly for low‑frequency components. Traditional compressed sensing guarantees rely on a small global coherence between sensing and sparsity dictionaries, a condition that fails when low‑order Fourier modes are highly correlated with low‑order Haar wavelets. To overcome this, the authors introduce the notion of local coherence, which quantifies the correlation of each individual sensing vector with the sparsity basis rather than the worst‑case across all vectors.

For the specific pair of Fourier measurements and Haar wavelet sparsity, they derive an explicit bound on the local coherence μ_k of the k‑th Fourier frequency: μ_k ≤ C/(1+|k|)^2. This bound shows that high‑frequency Fourier samples are nearly orthogonal to the wavelet basis, while low frequencies exhibit larger coherence. Leveraging this structure, the authors propose a variable‑density sampling scheme where each frequency is selected with probability proportional to the square of its local coherence. Practically, this translates to sampling frequencies from an inverse‑square power‑law distribution p(k) ∝ 1/(1+|k|)^2.

The main theoretical contribution is a proof that, under this sampling distribution, a measurement matrix formed from m randomly drawn Fourier rows satisfies the Restricted Isometry Property (RIP) of order s with high probability, provided

  m = O(s · log⁴ N),

where N is the ambient dimension and s the sparsity level (or total variation sparsity). Crucially, the proof only requires a bounded average coherence rather than a bounded maximum coherence, thereby relaxing the classical incoherence requirement.

With RIP established, the paper derives robust recovery guarantees for two standard reconstruction models:

1. ℓ₁‑minimization in the wavelet domain, solving min‖Ψx‖₁ subject to ‖Ax – y‖₂ ≤ ε.
2. Total Variation (TV) minimization, solving min‖∇x‖₁ subject to the same data‑fidelity constraint.

For both models, the reconstruction error obeys

  ‖x̂ – x‖₂ ≤ C₁·(σ/√m) + C₂·(‖x – x_s‖₁/√s)

or, in the TV case,

  ‖x̂ – x‖₂ ≤ C₃·(σ/√m) + C₄·(TV(x) – TV(x_s))/√s,

where σ denotes measurement noise level, x_s is the best s‑sparse approximation, and C_i are absolute constants. These bounds demonstrate stability to noise and robustness to sparsity defects, matching the best known guarantees for incoherent sensing scenarios.

Empirical evaluations on standard test images and on MRI data confirm the theory. Compared with uniform random sampling and with earlier power‑law sampling strategies, the proposed local‑coherence‑driven scheme yields higher PSNR (typically 3–5 dB improvement) and better structural similarity, especially for images dominated by low‑frequency content. The error curves as a function of noise level follow the predicted linear trend, corroborating the derived error bounds.

Beyond the specific Fourier–Haar pair, the authors argue that the local coherence framework is universally applicable. Whenever the average coherence between a sensing basis and a sparsity basis can be bounded, one can design a sampling distribution proportional to the squared local coherence to achieve near‑optimal RIP guarantees. This insight opens avenues for compressed sensing with other structured measurements (e.g., chirp, random convolutions) and for hybrid modalities that combine multiple sensing operators.

In conclusion, the paper makes three pivotal contributions: (i) it formalizes local coherence as a refined metric for sensing–sparsity interactions; (ii) it shows that an inverse‑square variable‑density sampling pattern yields near‑optimal RIP and stable recovery for Fourier‑Haar systems; and (iii) it generalizes the principle that average rather than worst‑case coherence suffices for optimal sparse recovery, provided the sampling strategy is appropriately adapted. These results bridge a long‑standing theoretical gap and provide a practical blueprint for designing sampling patterns in real‑world compressive imaging systems such as MRI, computed tomography, and remote sensing.