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 1 optimization problem, and the goal of the paper is to quantify the resolution. We consider both narrow-band and broadband 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.

Acousto-electric tomography is a hybrid imaging technique that aims to overcome the ill-posedness of the electric impedance tomography. We consider the problem of reconstructing the internal conductivity of an object by making electric measurements on the boundary while perturbing the conductivity by sending ultrasound waves to the object. We show that the conductivity can be uniquely recovered by using one boundary potential. AET is reduced to an inverse problem with internal data, and corresponding uniqueness and Lipschitz-type stability results are given. An iterative method for reconstructing the current and then the conductivity is presented along with numerical examples.

In this paper, we consider invisibility cloaking via the transformation optics approach through a ‘blow-up’ construction. An ideal cloak makes use of singular cloaking material. ‘Blow-up-a-small-region’ construction and ‘truncation-of-singularity’ construction are introduced to avoid the singular structure, however, giving only near-cloaks. The study in the literature is to develop various mechanisms in order to achieve high-accuracy approximate near-cloaking devices, and also from a practical viewpoint to nearly cloak an arbitrary content. We study the problem from a different viewpoint. It is shown that for those regularized cloaking devices, the corresponding scattering wave fields due to an incident plane wave have regular patterns. The regular patterns are both a curse and a blessing. On the one hand, the regular wave pattern betrays the location of a cloaking device which is an intrinsic defect due to the ‘blow-up’ construction, and this is particularly the case for the construction by employing a high-loss layer lining. Indeed, our numerical experiments show robust reconstructions of the location, even by implementing the phaseless cross-section data. The construction by employing a high-density layer lining shows a certain promising feature. On the other hand, it is shown that one can introduce an internal point source to produce the canceling scattering pattern to achieve a near-cloak of an arbitrary order of accuracy.

Abstract. 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.Key words. waves in random media, coherent interferometric imaging, l 1 optimization, mutual coherence.1. Introduction. Waves measured by a collection of nearby sensors, called an array of receivers, carry information about their source and the medium through which they travel. We consider a typical remote sensing regime with sources of small (point-like) support, and study the inverse problem of determining them from the array measurements.When the waves travel in a known and non-scattering (e.g. homogeneous) medium, the sources can be localized with reverse time migration [4,5] also known as backprojection [20]. This estimates the source locations as the peaks of the image formed by superposing the array recordings delayed by travel times from the receivers to the imaging points. The accuracy of the estimates depends on the array aperture, the distance of the sources from the array, and the temporal support of the signals emitted by the sources. It may be improved under certain conditions by using l 1 optimization, which seeks to invert the linear mapping from supposedly sparse vectors of the discretized source amplitude on some mesh, to the array measurements. The fast growing literature of imaging with l 1 optimization in homogeneous media includes compressed sensing studies such as [19,18], synthetic radar imaging studies like [1,8], array imaging studies like [14], and the resolution study [7].In this paper we assume that the waves travel in heterogeneous media with fluctuations of the wave speed caused by numerous inhomogeneities. The amplitude of the fluctuations is small, meaning that a single inhomogeneity is a weak scatterer. However, there are many inhomogeneities that interact with the waves on their way from the sources to the receivers, and their scattering effect accumulates. Because in appli...

Suppose that the set T = {T 1 , T 2 , . . . , T q } of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d 1 , . . . , d q } ⊂ R n , there exists a unique non-empty compact, which is typically a fractal set. We use the infinite digit expansions of the points of F to give simple necessary and sufficient conditions for the convex hull of F to be a polytope. Additionally, we present a technique to determine the vertices of such polytopes. These answer some of the related questions of Strichartz and Wang, and also enable us to approximate the Lebesgue measure of such self-affine sets. To show the use of our results, we also give several examples including the Levy dragon and the Heighway dragon.