Compressive Imaging: Stable and Robust Recovery from Variable Density Frequency Samples

To date, the theory for compressive sampling with frequency measurements has only been developed for bases that are incoherent with the Fourier basis. In many applications, such as Magnetic Resonance Imaging (MRI) or inverse scattering, one instead acquires images that are sparse in transform domains such as spatial finite differences or wavelets which are not incoherent with the Fourier basis. For these applications, overwhelming empirical evidence and heuristic arguments have suggested that superior image reconstruction can be obtained through certain variable density sampling strategies which concentrate on lower frequencies. In this talk we discuss recent theoretical reconstruction guarantees to corroborate these empirical results, showing that sampling frequencies according to suitable power-law densities enables image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results hinge on proving that the coherence between the Fourier and Haar wavelet basis is sufficiently concentrated on low frequencies that an incoherent preconditioned system results by resampling the Fourier basis appropriately. We finish by discussing the application of resampling coherent systems towards solving certain classical problems in approximation theory.