Liliana Borcea

Electrical impedance tomography

We review theoretical and numerical studies of the inverse problem of electrical impedance tomography which seeks the electrical conductivity and permittivity inside a body, given simultaneous measurements of electrical currents and potentials at the boundary.

Imaging and time reversal in random media

We present a general method for estimating the location of small, well-separated scatterers in a randomly inhomogeneous environment using an active sensor array. The main features of this method are (i) an arrival time analysis (ATA) of the echo received from the scatterers, (ii) a singular value decomposition of the array response matrix in the frequency domain, and (iii) the construction of an objective function in the time domain that is statistically stable and peaks on the scatterers. By statistically stable we mean here that the objective function is self-averaging over individual realizations of the medium. This is a new approach to array imaging that is motivated by time reversal in random media, analysed in detail previously. It combines features from seismic imaging, like ATA, with frequency-domain signal subspace methodology like multiple signal classification. We illustrate the theory with numerical simulations for ultrasound.

On the magneto-elastic properties of elastomer–ferromagnet composites

Abstract We study the macroscopic magneto-mechanical behavior of composite materials consisting of a random, statistically homogeneous distribution of ferromagnetic, rigid inclusions embedded firmly in a non-magnetic elastic matrix. Specifically, for given applied elastic and magnetic fields, we calculate the overall deformation and stress–strain relation for such a composite, correct to second order in the particle volume fraction. Our solution accounts for the fully coupled magneto-elastic interactions; the distribution of magnetization in the composite is calculated from the basic minimum energy principle of magneto-elasticity.

Theory and applications of time reversal and interferometric imaging

In time reversal, an array of transducers receives the signal emitted by a localized source, time reverses it and re-emits it into the medium. The emitted waves back-propagate to the source and tend to focus near it. In a homogeneous medium, the cross-range resolution of the refocused field at the source location is λ0L/a, where λ0 is the carrier wavelength, L is the range and a is the array aperture. The refocusing spot size in a homogeneous medium is independent of the bandwidth of the pulse, but broad-band can help in reducing spurious Fresnel zones. In a noisy (random) medium, the cross-range resolution is improved beyond the homogeneous diffraction limit because the array can capture waves that move away from it at the source, but get scattered onto it by the inhomogeneities. We refer to this phenomenon as super-resolution of the time reversal process in random media. Super-resolution implies in particular that, because of multipathing, the array appears to have an effective aperture ae that is greater than a. Since ae depends on the scattering medium, it is not known. In this paper we present a brief review of time reversal theory in a remote sensing regime and a robust procedure for estimating ae from the signals received at the array. Knowing ae permits assessing quantitatively super-resolution in time reversal for applications in spatially localized communications with reduced interference. We also review interferometric imaging and its relation to time reversal and to matched field imaging. We show that ae quantifies in an explicit way the loss of resolution in interferometric array imaging.

Interferometric array imaging in clutter

We introduce a space-time interferometric array imaging functional that provides statistically stable images in cluttered environments. We also present a resolution theory for this imaging functional that relates the space-time coherence of the data to the range and cross-range resolution of the image. Extensive numerical simulations illustrate the theory and address some implementation issues.

Adaptive interferometric imaging in clutter and optimal illumination

A frequently used broadband array imaging method is Kirchhoff or travel time migration. In smooth and known media Kirchhoff migration works quite well, with range resolution proportional to the reciprocal of the bandwidth and cross range resolution that is proportional to the reciprocal of the array size. In a randomly inhomogeneous medium, Kirchhoff migration is unreliable because the images depend on the detailed scattering properties of the random medium that are not known. In Borcea et al (2005 Interferometric array imaging in clutter Inverse Problems 21 1419–60) we introduced an imaging functional that does not depend on the detailed properties of the random medium, that is, it is statistically stable. This is the coherent interferometric (CINT) imaging functional, which can be viewed as a smoothed version of Kirchhoff migration. Smoothing increases the statistical stability of the image but causes blurring. In this paper, we introduce an adaptive version of CINT in which there is an optimal trade-off between statistical stability and blurring. We also introduce optimal illumination schemes for achieving the best possible resolution of the images obtained with CINT.

High-contrast impedance tomography

We introduce an output least-squares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modelled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate the performance of the method.

Statistically stable ultrasonic imaging in random media

Analysis of array data from acoustic scattering in a random medium with a small number of isolated targets is performed in order to image and, thereby, localize the spatial position of each target. Because the host medium has random fluctuations in wave speed, the background medium is itself a source of scattered energy. It is assumed, however, that the targets are sufficiently larger and/or more reflective than the background fluctuations so that a clear distinction can be made between targets and background scatterers. In numerical simulations nonreflective boundary conditions are used so as to isolate the effects of the host randomness from those of the spatial boundaries, which can then be treated in a separate analysis. It is shown that the key to successful imaging is finding statistically stable functionals of the data whose extreme values provide estimates of scatterer locations. The best ones are related to the eigenfunctions and eigenvalues of the array response matrix, just as one might expect from prior work on array data processing in complex scattering media having homogeneous backgrounds. The specific imaging functionals studied include matched-field processing and linear subspace methods, such as MUSIC (MUtiple SIgnal Classification). But statistical stability is not characteristic of the frequency domain, which is often the province of these methods. By transforming back into the time domain after first diagonalizing the array data in the frequency domain, one can take advantage of both the time-domain stability and the frequency-domain orthogonality of the relevant eigenfunctions.

Coherent interferometric imaging in clutter

Coherent interferometry is an array-imaging method in which we back-propagate, or migrate, crosscorrelations of the traces over appropriately chosen space-time windows rather than the traces themselves. The size of the space-time windows is critical and depends on two parameters. One is the decoherence frequency, which is proportional to the reciprocalofthedelayspreadinthetracesproducedbytheclutter. The other is the decoherence length, which also depends on the clutter. As is usual, the clutter is modeled by random fluctuationsinthemediumproperties.Inisotropicclutter,the decoherence length is typically much shorter than the array aperture. In layered random media, the decoherence length along the layers can be quite long. We show that when the crosscorrelations of the traces are calculated adaptively, coherentinterferometrycanprovideimagesthatarestatistically stable relative to small-scale clutter in the environment.This means that the images we obtain are not sensitive to the detailedformoftheclutter.Theyonlydependonitsoverallstatistical properties. However, clutter does reduce the resolution of the images by blurring. We show how the amount of blurring can be minimized by using adaptive interferometric imaging algorithms and discuss the relation between the coherence properties of the array data and the loss in resolution causedbytheblurring.

A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency

We propose a nonlinear multigrid approach for imaging the electrical conductivity and permittivity of a body Ω, given partial, usually noisy knowledge of the Neumann-to-Dirichlet map at the boundary. The algorithm is a nested iteration, where the image is constructed on a sequence of grids in Ω, starting from the coarsest grid and advancing towards the finest one. We show various numerical examples that demonstrate the effectiveness and robustness of the algorithm and prove local convergence.