Abstract
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.
Highlights
Many problems in science and engineering require solving an inverse problem, where parameters of interest, x, are estimated from a set of linear measurements, y
We presented Element based compressive sensing (CS) theory that can be applicable to any sensing matrix
We showed that the sparse recovery would be exact, if: 1) (AK AK)−1 exists, and
Summary
Many problems in science and engineering require solving an inverse problem, where parameters of interest, x, are estimated from a set of linear measurements, y. Transform [1], i.i.d entries from a Gaussian distribution with zero mean [14], and from an orthogonal matrix [15] For these matrices, it has been shown that if the number of measurements, m is about the order of O(s log n), where n is the number of unknowns, the RIP holds. The approach relies on the sensing matrix confining local isotropy and incoherence (μ) properties that are shown to be valid for orthogonal matrices This approach is extended to anisotropic measurements in [18] (anisotropic measurements are defined as AΩ AΩ = I) with an additional condition that AΩ AΩ is invertible, where AΩ is a superset of measurements from which m measurements are chosen in A.
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