Ilker Kocyigit

Liliana Borcea, Ilker Kocyigit

We study an inverse problem for the wave equation where localized wave sources in random scattering media are to be determined from time resolved measurements of the waves at an array of receivers. The sources are far from the array, so the measurements are affected by cumulative scattering in the medium, but they are not further than a transport mean free path, which is the length scale characteristic of the onset of wave diffusion that prohibits coherent imaging. The inversion is based on the Coherent Interferometric (CINT) imaging method which mitigates the scattering effects by introducing an appropriate smoothing operation in the image formation. This smoothing stabilizes statistically the images, at the expense of their resolution. We complement the CINT method with a convex ($l_1$) optimization in order to improve the source localization and obtain quantitative estimates of the source intensities. We analyze the method in a regime where scattering can be modeled by large random wavefront distortions, and quantify the accuracy of the inversion in terms of the spatial separation of individual sources or clusters of sources. The theoretical predictions are demonstrated with numerical simulations.

Liliana Borcea, Ilker Kocyigit

We present a novel algorithm for high resolution coherent imaging of sound sources in random scattering media using time resolved measurements of the acoustic pressure at an array of receivers. The sound waves travel a long distance between the sources and receivers so that they are significantly affected by scattering in the random medium. We model the scattering effects by large random wavefront distortions, but the results extend to stronger effects, as long as the waves retain some coherence i.e., before the onset of wave diffusion. It is known that scattering in random media can be mitigated in imaging using coherent interferometry (CINT). This method introduces a statistical stabilization in the image formation, at the cost of image blur. We show how to modify the CINT method in order to image wave sources that are too close to each other to be distinguished by CINT alone. We introduce the algorithm from first principles and demonstrate its performance with numerical simulations.

Liliana Borcea, Ilker Kocyigit

We study array imaging of a sparse scene of point-like sources or scatterers in a homogeneous medium. For source imaging the sensors in the array are receivers that collect measurements of the wave field. For imaging scatterers the array probes the medium with waves and records the echoes. In either case the image formation is stated as a sparsity promoting $\ell_1$ optimization problem, and the goal of the paper is to quantify the resolution. We consider both narrow-band and broad-band imaging, and a geometric setup with a small array. We take first the case of the unknowns lying on the imaging grid, and derive resolution limits that depend on the sparsity of the scene. Then we consider the general case with the unknowns at arbitrary locations. The analysis is based on estimates of the cumulative mutual coherence and a related concept, which we call interaction coefficient. It complements recent results in compressed sensing by deriving deterministic resolution limits that account for worse case scenarios in terms of locations of the unknowns in the imaging region, and also by interpreting the results in some cases where uniqueness of the solution does not hold. We demonstrate the theoretical predictions with numerical simulations.