Ilker Kocyigit

Acousto-electric tomography and CGO solutions with internal data

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.

Resolution Analysis of Imaging with

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

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Optimization

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 the interaction coefficient. The latter complements recent results in compressed sensing by deriving deterministic resolution limits that account for worst-...

REMARKS ON SELF-AFFINE FRACTALS WITH POLYTOPE CONVEX HULLS

Suppose that the set of real n × n matrices has joint spectral radius less than 1. Then for any digit set D = {d1, …, dq} ⊂ ℝn, there exists a unique non-empty compact set satisfying , 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.

Imaging in Random Media with Convex Optimization

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 the images statistically, at the expense of their resolution. We complement the CINT method with a convex (

On the Dimension of Self-Affine Fractals

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A Multiple Measurement Vector Approach to Synthetic Aperture Radar Imaging

) 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 distortio...

Regular scattering patterns from near-cloaking devices and their implications for invisibility cloaking

An n ×n matrix M is called expanding if all its eigenvalues have moduli > 1. Let A be a nonempty finite set of vectors in the n-dimensional Euclidean space. Then there exists a unique nonempty compact set F satisfying \(MF\,=\,F + A\). F is called a self-affine set or a self-affine fractal. F can also be considered as the attractor of an affine iterated function system. Although such sets are basic structures in the theory of fractals, there are still many problems on them to be studied. Among those problems, the calculation or the estimation of fractal dimensions of F is of considerable interest. In this work, we discuss some problems about the singular value dimension of self-affine sets. We then generalize the singular value dimension to certain graph directed sets and give a result on the computation of it. On the other hand, for a very few classes of self-affine fractals, the Hausdorff dimension and the singular value dimension are known to be different. Such fractals are called exceptional self-affine fractals. Finally, we present a new class of exceptional self-affine fractals and show that the generalized singular value dimension of F in that class is the same as the box (counting) dimension.

On the Convex Hulls of Self-Affine Fractals

We study a multiple measurement vector (MMV) approach to synthetic aperture radar (SAR) imaging of scenes with direction dependent reflectivity and with polarization diverse measurements. The data are gathered by a moving transmit- receive platform which probes the imaging scene with signals and records the backscattered waves. The unknown reflectivity is represented by a matrix with row support corresponding to the location of the scatterers in the scene, and columns corresponding to measurements gathered from different sub-apertures, or different polarization of the waves. The MMV methodology is used to estimate the reflectivity matrix by inverting in an appropriate sense the linear system of equations that models the SAR data. We obtain a resolution analysis of SAR imaging with MMV, which takes into account the sparsity of the imaging scene, the separation of the scatterers and the diversity of the measurements. The results of the analysis are illustrated with some numerical simulations.