Matrix Element-Based Theory of Compressive Sensing and Its Application to Electromagnetic Imaging

We present a simple and general theory of compressive sensing (CS) that relies on elements of the sensing matrix rather than on the number of measurements. We prove the exact recovery using a dual certificate by showing that the sensing matrix satisfies an incoherence property and isotropy property if the sparsity level is kept lower than the reciprocal of the largest element of a matrix created from the sensing matrix. Unlike the CS literature, this unconventional approach does not require a linear relationship between the sparsity and the number of measurements and at the same time, can easily be evaluated. This adaptability captures anisotropic measurements appropriately as with anisotropic measurements, adding more measurements does not really imply that a signal with more nonzero elements will be recovered exactly. As an illustration, we demonstrate the theory’s ability to accurately handle the anisotropic (Green’s function-based sensing matrix) measurements and also its similarity to the existing CS literature for isotropic (Fourier) measurements. Further, we show the usefulness of the theory in comparing different sensing matrices and in generating dielectric images. The dielectric images are perfectly recovered even when there is only a single transmitter.