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Electrical Impedance Tomography
 SIAM REVIEW
, 1999
"... This paper surveys some of the work our group has done in electrical impedance tomography. ..."
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Cited by 86 (2 self)
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This paper surveys some of the work our group has done in electrical impedance tomography.
RESOLUTION AND STABILITY ANALYSIS OF AN INVERSE PROBLEM IN ELECTRICAL IMPEDANCE TOMOGRAPHY  DEPENDENCE ON THE INPUT CURRENT PATTERNS
"... Electrical impedance tomography is a procedure by which one nds the conductivity distribution inside a domain from measurements of voltages and currents at the boundary. This work addresses the issue of stability and resolution limit of such an imaging device. We consider the realistic case where o ..."
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Cited by 14 (1 self)
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Electrical impedance tomography is a procedure by which one nds the conductivity distribution inside a domain from measurements of voltages and currents at the boundary. This work addresses the issue of stability and resolution limit of such an imaging device. We consider the realistic case where only a nite number of measurements are available. An important feature of our approach, which is based on linearization, is that we do not discretize the unknown conductivity distribution. Instead, we de ne a pseudosolution based on leastsquares. A goal of this investigation is to compare the stability and resolution power of a system that uses dipole sources, with another that uses trigonometric sources. Our ndings are illustrated in numerical calculations.
Multiscale, statistical anomaly detection analysis and algorithms for linearized inverse scattering problems
 Multidimensional Systems and Signal Processing
, 1997
"... Abstract. In this paper we explore the utility of multiscale and statistical techniques for detecting and characterizing the structure of localized anomalies in a medium based upon observations of scattered energy obtained at the boundaries of the region of interest. Wavelet transform techniques are ..."
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Cited by 9 (5 self)
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Abstract. In this paper we explore the utility of multiscale and statistical techniques for detecting and characterizing the structure of localized anomalies in a medium based upon observations of scattered energy obtained at the boundaries of the region of interest. Wavelet transform techniques are used to provide an efficient and physically meaningful method for modeling the nonanomalous structure of the medium under investigation. We employ decisiontheoretic methods both to analyze a variety of difficulties associated with the anomaly detection problem and as the basis for an algorithm to perform anomaly detection and estimation. These methods allow for a quantitative evaluation of the manner in which the performance of the algorithms is impacted by the amplitudes, spatial sizes, and positions of anomalous areas in the overall region of interest. Given the insight provided by this work, we formulate and analyze an algorithm for determining the number, location, and magnitudes associated with a set of anomaly structures. This approach is based upon the use of a Generalized, Mary Likelihood Ratio Test to successively subdivide the region as a means of localizing anomalous areas in both space and scale. Examples of our multiscale inversion algorithm are presented using the Born approximation of an electrical conductivity problem formulated so as to illustrate many of the features associated with similar detection problems arising in fields such as geophysical prospecting, ultrasonic imaging, and medical imaging. Key Words: 1.
Optimal acoustic measurements
 SIAM J. Appl. Math
"... We consider the problem of obtaining information about an inaccessible halfspace from acoustic measurements made in the accessible halfspace. If the measurements are of limited precision, some scatterers will be undetectable because their scattered fields are below the precision of the measuring i ..."
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Cited by 8 (2 self)
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We consider the problem of obtaining information about an inaccessible halfspace from acoustic measurements made in the accessible halfspace. If the measurements are of limited precision, some scatterers will be undetectable because their scattered fields are below the precision of the measuring instrument. How can we make measurements that are optimal for detecting the presence of an object? In other words, what incident fields should we apply that will result in the biggest measurements? There are many ways to formulate this question, depending on the measuring instruments. In this paper we consider a formulation involving wavesplitting in the accessible halfspace: what downgoing wave will result in an upgoing wave of greatest energy? A closely related question arises in the case when we have a guess about the configuration of the inaccessible halfspace. What measurements should we make to determine whether our guess is accurate? In this case we compare the scattered field to the field computed from the guessed configuration. Again we look for the incident field that results in the greatest energy difference. We show that the optimal incident field can be found by an iterative process involving time reversal “mirrors”. For bandlimited incident fields and compactly supported scatterers, in the generic case this iterative process converges to a single timeharmonic field. In particular, the process automatically “tunes ” to the best frequency. This analysis provides a theoretical foundation for the frequencyshifting and pulsebroadening observed in certain computations [3] and timereversal experiments [14] [15].
On the Parametrization of Illposed Inverse Problems Arising from Elliptic Partial Differential Equations
, 2006
"... ..."
Electrical conductivity imaging via contactless measurements
 IEEE TRANS. ON MED. IMAG
, 1999
"... A new imaging modality is introduced to image electrical conductivity of biological tissues via contactless measurements. This modality uses magnetic excitation to induce currents inside the body and measures the magnetic fields of the induced currents. In this study, the mathematical basis of the ..."
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Cited by 7 (2 self)
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A new imaging modality is introduced to image electrical conductivity of biological tissues via contactless measurements. This modality uses magnetic excitation to induce currents inside the body and measures the magnetic fields of the induced currents. In this study, the mathematical basis of the methodology is analyzed and numerical models are developed to simulate the imaging system. The induced currents are expressed using the � e0 formulation of the electric field where � e is the magnetic vector potential and 0 is the scalar potential function. It is assumed that � e describes the primary magnetic vector potential that exists in the absence of the body. This assumption considerably simplifies the solution of the secondary magnetic fields caused by induced currents. In order to solve 0 for objects of arbitrary conductivity distribution a threedimensional (3D) finiteelement method (FEM) formulation is employed. A specific 7 2 7coil system is assumed nearby the upper surface of a 10 2 10 2 5cm conductive body. A sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singularvalue decomposition of the sensitivity matrix shows various characteristics of the imaging system. Images are reconstructed using 500 voxels in the image domain, with truncated pseudoinverse. The noise level is assumed to produce a representative signaltonoise ratio (SNR) of 80 dB. It is observed that it is possible to identify voxel perturbations (of volume 1 cm Q) at 2 cm depth. However, resolution gradually decreases for deeper conductivity perturbations.
Circular resistor networks for electrical impedance tomography with partial boundary measurements.
"... Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in ..."
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Cited by 5 (4 self)
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Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on socalled optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting tradeoff between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary. 1.
Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements
"... ..."
Resistor network approaches to electrical impedance tomography
 Inside Out, Mathematical Sciences Research Institute Publications
, 2011
"... We review a resistor network approach to the numerical solution of the inverse problem of electrical impedance tomography (EIT). The networks arise in the context of finite volume discretizations of the elliptic equation for the electric potential, on sparse and adaptively refined grids that we call ..."
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Cited by 2 (1 self)
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We review a resistor network approach to the numerical solution of the inverse problem of electrical impedance tomography (EIT). The networks arise in the context of finite volume discretizations of the elliptic equation for the electric potential, on sparse and adaptively refined grids that we call optimal. The name refers to the fact that the grids give spectrally accurate approximations of the Dirichlet to Neumann map, the data in EIT. The fundamental feature of the optimal grids in inversion is that they connect the discrete inverse problem for resistor networks to the continuum EIT problem. 1.
Reconstructions of chest phantoms by the Dbar method for electrical impedance tomography
 IEEE Trans. Med. Imaging
, 2004
"... Abstract — The problem this paper addresses is how to use the 2D Dbar method for electrical impedance tomography with experimental data collected on finitely many electrodes covering a portion of the boundary of a body. This requires an approximation of the DirichlettoNeumann, or voltagetocurre ..."
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Cited by 2 (1 self)
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Abstract — The problem this paper addresses is how to use the 2D Dbar method for electrical impedance tomography with experimental data collected on finitely many electrodes covering a portion of the boundary of a body. This requires an approximation of the DirichlettoNeumann, or voltagetocurrent density map, defined on the entire boundary of the region, from a finite number of matrix elements of the currenttovoltage map. Reconstructions from experimental data collected on a saline filled tank containing agar heart and lung phantoms are presented, and the results are compared to reconstructions by the NOSER algorithm on the same data. Index Terms — electrical impedance tomography, direct reconstruction algorithm, Dbar method I.