Stable and robust sampling strategies for compressive imaging

TL;DR

This paper introduces a new sampling strategy based on local coherence for compressive imaging, enabling stable and robust image reconstruction from frequency samples, especially when sparsity is in wavelet domains.

Contribution

It develops a local coherence framework that improves sampling guarantees for Fourier and wavelet sparsity, leading to near-optimal compressed sensing results.

Findings

Proves restricted isometry property with variable-density sampling.

Demonstrates stability to sparsity defects and robustness to noise.

Applicable to both $\,\ell_1$-minimization and total variation minimization.

Abstract

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.