Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

Borcea, Liliana; Druskin, Vladimir; Mamonov, A.; Vasquez, Fernando Guevara

doi:10.1088/0266-5611/26/10/105009

Cited by 22 publications

(60 citation statements)

References 40 publications

(180 reference statements)

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“…Instead of relating operators L σ,0 and L 1,q , we can also consider operators L σ1,0 and L σ0,q , for two positive conductivities σ 0 and σ 1 and a potential q. It follows by straightforward calculations that (7) (…”

Section: Continuum Inverse Conductivity and Schrödinger Problemsmentioning

confidence: 99%

“…They are based on the circular planar resistor networks studied in [20,27] and are described briefly below. We refer to [5,7,6] and the review [8] for details on how to use the parametric reduced models to determine a discrete Laplacian which is consistent with the measurements of the DtN map in the continuum setting, and to recover the unknown conductivity.…”

Section: Discrete Inverse Conductivity and Schrödinger Problemsmentioning

confidence: 99%

“…Unlike the continuum case, where σ ≡ 1 corresponds to L 1,0 = ∆, the case of constant discrete conductivity γ = 1 has no special significance in the discrete setting. However, the reduced model (resistor network) for the reference conductivity σ o ≡ 1 (i.e., q ≡ 0) is a good approximation of the Laplacian on the graph G, as shown in [5,7,8]. The geometric averages of its edge conductivities γ 0 define the line graph Laplacian.…”

Section: Discrete Inverse Conductivity and Schrödinger Problemsmentioning

confidence: 99%

“…In this paper we show how to use the resistor networks with circular planar graphs, the reduced models for the inverse conductivity problem in [5,7,6,8], to solve the inverse Schrödinger problem with absorptive potentials in two dimensions. The inverse conductivity and Schrödinger problems are connected in the continuum by a well known Liouville identity [29].…”

Section: Introductionmentioning

confidence: 99%

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A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

Borcea¹,

Vasquez

Mamonov

2017

Inverse Problems &Amp; Imaging

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We propose a discrete approach for solving an inverse problem for Schrödinger's equation in two dimensions, where the unknown potential is to be determined from boundary measurements of the Dirichlet to Neumann map. For absorptive potentials, and in the continuum, it is known that by using the Liouville identity we obtain an inverse conductivity problem. Its discrete analogue is to find a resistor network that matches the measurements, and is well understood. Here we show how to use a discrete Liouville identity to transform its solution to that of Schrödinger's problem. The discrete Schrödinger potential given by the discrete Liouville identity can be used to reconstruct the potential in the continuum in two ways. First, we can obtain a direct but coarse reconstruction by interpreting the values of the discrete Schrödinger potential as averages of the continuum Schrödinger potential on a special sensitivity grid. Second, the discrete Schrödinger potential may be used to reformulate the conventional nonlinear output least squares optimization formulation of the inverse Schrödinger problem. Instead of minimizing the boundary measurement misfit, we minimize the misfit between the discrete Schrödinger potentials. This results in a better behaved optimization problem that converges in a single Gauss-Newton iteration, and gives good quality reconstructions of the potential, as illustrated by the numerical results.

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Section: Continuum Inverse Conductivity and Schrödinger Problemsmentioning

confidence: 99%

Section: Discrete Inverse Conductivity and Schrödinger Problemsmentioning

confidence: 99%

Section: Discrete Inverse Conductivity and Schrödinger Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

Borcea¹,

Vasquez

Mamonov

2017

Inverse Problems &Amp; Imaging

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show abstract

“…In [23,24] for instance, optimal grids are constructed via conformal mappings and solutions with minimum anisotropy are recovered. We instead provide a framework in which optimal grids are naturally identified via harmonic coordinates and weighted Delaunay triangulations; solutions are then naturally represented via convex functions, without introducing any bias on the possible anisotropy of the solutions.…”

Section: Divergence-free Parameterization Recoverymentioning

confidence: 99%

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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Overdetermined partial resolvent kernels for finite networks

Araúz

Carmona

Encinas

2016

Journal of Mathematical Analysis and Applications

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In [2], a study of the existence and uniqueness of solution of partial overdetermined boundary value problems for finite networks was performed. These problems involve Schrödinger operators and the novel feature is that no data are prescribed on part of the boundary, whereas both the values of the function and of its normal derivative are given on another part of the boundary. In the present work, we study the resolvent kernels associated with overdetermined partial boundary value problems on finite network and we express them in terms of the well-known Green operator and the Dirichlet-to-Robin map. Moreover, we analyze their main properties and we compute them in the case of a generalized cylinder. The obtained expression involve polynomials that can be seen as a generalization of Chebyshev polynomias, and indeed when the conductances along axes are constant the expressions for the overdetermined partial resolvent kernels are given in terms of second kind Chebyshev polynomials.

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Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

Cited by 22 publications

References 40 publications

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

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

Overdetermined partial resolvent kernels for finite networks

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