Beyond incoherence: 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 compressed 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, so for matrices comprised of frequencies sampled from suitable power-law densities, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategies we provide — which are independent of the ambient dimension up to logarithmic factors — allow for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by `1-minimization and by total variation minimization.