Margaret Cheney

Electrical Impedance Tomography

Electrical impedance tomography (EIT) is an imaging modality that estimates the electrical properties at the interior of an object from measurements made on its surface. Typically, currents are injected into the object through electrodes placed on its surface, and the resulting electrode voltages are measured. An appropriate set of current patterns, with each pattern specifying the value of the current for each electrode, is applied to the object, and a reconstruction algorithm uses knowledge of the applied current patterns and the measured electrode voltages to solve the inverse problem, computing the electrical conductivity and permittivity distributions in the object. This article focuses on the type of EIT called adaptive current tomography (ACT) in which currents are applied simultaneously to all the electrodes. A number of current patterns are applied, where each pattern defines the current for each electrode, and the subsequent electrode voltages are measured to generate the data required for image reconstruction. A ring of electrodes may be placed in a single plane around the object, to define a two-dimensional problem, or in several layers of such rings, to define a three-dimensional problem. The reconstruction problem is described and two algorithms are discussed, a one-step, two-dimensional (2-D) Newton-Raphson algorithm and a one-step, full three-dimensional (3-D) reconstructor. Results from experimental data are presented which illustrate the performance of the algorithms.

Existence and uniqueness for electrode models for electric current computed tomography

The following experiment is considered. To a body of given conductivity, a number of electrodes are attached, through which current is sent. On the same electrodes, the resulting voltages are measured. This experiment can be described by a number of mathematical models [K.-S. Cheng et al., IEEE Transactions on Biomedical Engineering, 36 (1989), pp. 918–924]. These models are discussed and their predictions compared with experiment. In particular, a model is exhibited that is capable of predicting the experimentally measured voltages to within 0.1 percent. For this model, existence and uniqueness of the associated electrical potential is proved.

NOSER: An algorithm for solving the inverse conductivity problem

The inverse conductivity problem is the mathematical problem that must be solved in order for electrical impedance tomography systems to be able to make images. Here we show how this inverse conductivity problem is related to a number of other inverse problem. We then explain the workings of an algorithm that we have used to make images from electrical impedance data measured on the boundary of a circle in two dimensions. This algorithm is based on the method of least squares. It takes one step of a Newtons method, using a constant conductivity as an initial guess. Most of the calculations can therefore be done analytically. The resulting code is named NOSER, for Newtons One‐Step Error Reconstructor. It provides a reconstruction with 496 degrees of freedom. The code does not reproduce the conductivity accurately (unless it differs very little from a constant), but it yields useful images. This is illustrated by images reconstructed from numerical and experimental data, including data from a human chest.

The linear sampling method and the MUSIC algorithm

This paper gives a short tutorial on the MUSIC algorithm (Devaney, Therrien) and the linear sampling method of Kirsch, and explains how the latter is an extension of the former. In particular, for the case of scattering from a finite number of weakly scattering targets, the two algorithms are identical.

Fundamentals of radar imaging

List of figures List of tables Preface Part I. Radar Basic: 1. Introduction 2. Radar systems 3. Introduction to scattering 4. Detection of signals in noise 5. The radar ambiguity function Part II. Radar Imaging: 6. Wave propagation in two and three dimensions 7. Inverse synthetic-aperture radar 8. Antennas 9. Synthetic-aperture radar 10. Related techniques 11. Open problems Bibliography Index.

A Mathematical Tutorial on Synthetic Aperture Radar

This paper presents the foundations of conventional strip-mode synthetic aperture radar (SAR) from a mathematical point of view. In particular, the paper shows how a simple antenna model can be used together with a linearized scattering approximation to predict the received signal. The conventional matched-filter processing is explained and analyzed to exhibit the resolution of the SAR system.

Synthetic aperture inversion

This paper considers synthetic aperture radar and other synthetic aperture imaging systems in which a backscattered wave is measured from a variety of locations. The paper begins with a (linearized) mathematical model, based on the wave equation, that includes the effects of limited bandwidth and the antenna beam pattern. The model includes antennas with poor directionality, such as are needed in the problem of foliage-penetrating radar, and can also accommodate other effects such as antenna motion and steering. For this mathematical model, we use the tools of microlocal analysis to develop and analyse a three-dimensional imaging algorithm that applies to measurements made on a two-dimensional surface. The analysis shows that simple backprojection should result in an image of the singularities in the scattering region. This image can be improved by following the backprojection with a spatially variable filter that includes not only the antenna beam pattern and source waveform but also a certain geometrical scaling factor called the Beylkin determinant. Moreover, we show how to combine the backprojection and filtering in one step. The resulting algorithm places singularities in the correct locations, with the correct orientations and strengths. The algorithm is analysed to determine which information about the scattering region is reconstructed and to determine the resolution. We introduce a notion of directional resolution to treat the reconstruction of walls and other directional elements. We also determine the fineness with which the data must be sampled in order for the theoretical analysis to apply. Finally, we relate the present analysis to previous work and discuss briefly implications for the case of a single flight track.

Layer stripping: a direct numerical method for impedance imaging

An impedance imaging problem is to find the electrical conductivity and permittivity distributions inside a body from measurements made on the boundary. The following experiment is considered: a set of electric currents are applied to the surface of the body and the resulting voltages are measured on that surface. The authors describe the performance of a direct numerical method for approximating the conductivity in the interior. The algorithm proceeds via two steps: first the conductivity is found near the bounding surface of the body from the data having the highest available spatial frequency; next the boundary data on an interior surface are synthesized using a nonlinear differential equation of Riccati type. The process is then repeated, and an estimate of the conductivity is found, layer by layer. They establish the theoretical basis for the algorithm and report on numerical tests.

Distinguishability in impedance imaging

Impedance imaging systems apply currents to the surface of a body, measure the induced voltages on the surface, and from this information reconstruct an approximation to the electrical conductivity in the interior. A detailed discussion of several ways to measure the ability of such a system to distinguish between two different conductivity distributions is given. The subtle differences between these related measures are discussed, and examples are provided to show that these different measures can give rise to different answers to various practical questions about system design.<<ETX>>

Inverse problems for a perturbed dissipative half-space

Addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative half-space. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related to Maxwells equations. Section 2 introduces three formulations for direct and inverse problems for the half-space geometry. Two of these formulations relate to scattering problems, and the third to a boundary value problem. Section 3 shows how the scattering problems can be related to the boundary value problem. This shows that the three inverse problems are equivalent in a certain sense. In section 4, the boundary value problem is used to outline a simple way to formulate a multi-dimensional layer stripping procedure. This procedure is unstable and does not constitute a practical algorithm for solving the inverse problem. The paper concludes with three appendices, the first two of which carry out a careful construction of solutions of the direct problems and the third of which contains a discussion of some properties of the scattering operator.