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Bayesian compressive sensing for ultrawideband inverse scattering in random media

We develop an ultrawideband (UWB) inverse scattering technique for reconstructing continuous random media based on Bayesian compressive sensing. In addition to providing maximum a posteriori estimates of the unknown weights, Bayesian inversion provides estimate of the confidence level of the solution, as well as a systematic approach for optimizing subsequent measurement(s) to maximize information gain. We impose sparsity priors directly on spatial harmonics to exploit the spatial correlation exhibited by continuous media, and solve for their posterior probability density functions efficiently using a fast relevance vector machine. We linearize the problem using the first-order Born approximation which enables us to combine, in a single inversion, measurements from multiple transmitters and ultrawideband frequencies. We extend the method to high-contrast media using the distorted-Born iterative method. We apply time-reversal strategies to adaptively focus the inversion effort onto subdomains of interest, and hence reduce the overall inversion cost. The proposed techniques are illustrated in a number of canonical scenarios including crosshole and borehole sensing.