D‐Bar Method for Electrical Impedance Tomography with Discontinuous Conductivities

Knudsen, Kim; Lassas, Matti; Mueller, Jennifer L.; Siltanen, Samuli

doi:10.1137/060656930

Cited by 74 publications

(76 citation statements)

References 31 publications

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“…In [8] Calderón solved the linearized problem and proposed a linearized reconstruction algorithm. (See [10,26,27,16] for other work on Calderón's method.) Then in [17,18] Kohn and Vogelius proved uniqueness for piecewise real-analytic conductivities.…”

Section: Introductionmentioning

confidence: 99%

“…Since solving (3) is severely ill-posed, in [22] a simple approximation for the scattering transform is introduced in which we replace ψ| ∂Ω by e ikx | ∂Ω . This approximation denoted by t exp was studied further in [16] and used on experimental data in [11,12]. The ill-posedness of the inverse problem is manifested in the computation of the scattering transform.…”

Section: Introductionmentioning

confidence: 99%

“…The ill-posedness of the inverse problem is manifested in the computation of the scattering transform. Calderón's linearization method can be seen as an approximation of the D-bar method via three linearizations [16]; see Section 2.4 for details. In contrast, the implementation of the D-bar method with the t exp approximation only makes one linearizing assumption in the approximation of the scattering transform.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast, the implementation of the D-bar method with the t exp approximation only makes one linearizing assumption in the approximation of the scattering transform. Reconstructions of a radially symmetric piecewise constant conductivity with a single jump discontinuity were computed in [16] by both the D-bar method with the t exp approximation and by Calderón's method using explicit reconstruction formulas. The reconstructions were very similar in terms of the reconstructed current amplitude and their qualitative features.…”

Section: Introductionmentioning

confidence: 99%

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Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography

Knudsen

Lassas²,

Mueller

et al. 2008

J. Phys.: Conf. Ser.

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Abstract. The importance of the solution of the boundary integral equation for the exponentially growing solutions to the Schrödinger equation arising from the 2-D inverse conductivity problems is demonstrated by a study of reconstructions of simple piecewise constant conductivities on a disk from two methods of approximating the scattering transform in the Dbar method and from Calderón's linearization method. IntroductionThe inverse conductivity problem was posed by Calderón in the seminal paper [8] and it concerns the unique determination and reconstruction of an isotropic conductivity distribution in a bounded domain from electrostatic measurements on the boundary of the domain. This problem is the mathematical problem behind the technology for medical imaging known as electrical impedance tomography (EIT) (see [9,5] for review articles on EIT). Conductivity distributions appearing in applications are typically piecewise continuous. This is the case for example in medical EIT, since various tissues in the body have different conductivities with discontinuities at organ boundaries. Here we consider 2-D reconstructions. These can be used to image cross-sections of a 3-D region, such as a patient's torso. In the case of patients receiving mechanical ventilation, for example, 2-D cross-sections are useful for obtaining regional ventilation information in the lungs, which is valuable for setting and controlling the airflow and pressure settings on the ventilator [25,1]. Real-time imaging of cross-sectional lung activity can also be used for diagnostic purposes, such as detecting a lung collapse, a pulmonary embolism, pulmonary edema, or a pneumothorax.To state the problem formally, let Ω ⊂ R 2 be a bounded domain with smooth boundary ∂Ω and let γ ∈ L ∞ (Ω) be the conductivity distribution in Ω satisfying

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography

Knudsen

Lassas²,

Mueller

et al. 2008

J. Phys.: Conf. Ser.

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“…The 2D Calderón problem was solved for smooth conductivities in [8], and the algorithm was successfully implemented [9][10][11][12][13][14]. Several ideas from these implementations were used recently in a crude simplification of the reconstruction method for 3D [15,16].…”

Section: Introductionmentioning

confidence: 99%

Electrical impedance tomography: 3D reconstructions using scattering transforms

Delbary

Hansen

Knudsen

2012

Applicable Analysis

Self Cite

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In three dimensions the Calderón problem was addressed and solved in theory in the 1980s. The main ingredients in the solution of the problem are complex geometrical optics solutions to the conductivity equation and a (non-physical) scattering transform. The resulting reconstruction algorithm is in principle direct and addresses the full non-linear problem immediately. In this paper a new simplification of the algorithm is suggested. The method is based on solving a boundary integral equation for the complex geometrical optics solutions, and the method is implemented numerically using a Nyström method. Convergence estimates are obtained using hyperinterpolation operators. We compare the method numerically to two other approximations by evaluation on two numerical examples. In addition a moment method for the numerical solution of the forward problem is given.

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Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization

Desbrun

Donaldson

Owhadi

2013

Multiscale Analysis and Nonlinear Dynamics

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Imaging and simulation methods are typically constrained to resolutions much coarser than the scale of physical micro-structures present in body tissues or geological features. Mathematical and numerical homogenization address this practical issue by identifying and computing appropriate spatial averages that result in accuracy and consistency between the macro-scales we observe and the underlying micro-scale models we assume. Among the various applications benefiting from homogenization, Electric Impedance Tomography (EIT) images the electrical conductivity of a body by measuring electrical potentials consequential to electric currents applied to the exterior of the body. EIT is routinely used in breast cancer detection and cardio-pulmonary imaging, where current flow in fine-scale tissues underlies the resulting coarse-scale images.In this paper, we introduce a geometric approach for the homogenization (simulation) and inverse homogenization (imaging) of divergenceform elliptic operators with rough conductivity coefficients in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions over the domain. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of the domain when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. We explicitly give the transformations which map conductivity coefficients into divergencefree matrices and convex functions, as well as their respective inverses. Using weighted Delaunay triangulations for linearly interpolating convex functions, we apply this geometric framework to obtain a robust homogenization algorithm for arbitrary rough coefficients, extending the global optimality of Delaunay triangulations with respect to a discrete Dirichlet energy to weighted Delaunay triangulations. We then consider inverse homogenization, which is known to be both non-linear and severely illposed, but that we can decompose into a linear ill-posed problem and a well-posed non-linear problem. Finally, our new geometric approach to homogenization and inverse homogenization is applied to EIT.

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D‐Bar Method for Electrical Impedance Tomography with Discontinuous Conductivities

Cited by 74 publications

References 31 publications

Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography

Reconstructions of piecewise constant conductivities by the D-bar method for electrical impedance tomography

Electrical impedance tomography: 3D reconstructions using scattering transforms

Modeling Across Scales: Discrete Geometric Structures in Homogenization and Inverse Homogenization

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