Circular resistor networks for electrical impedance tomography with partial boundary measurements

Borcea, Liliana; Druskin, Vladimir; Mamonov, A.

doi:10.1088/0266-5611/26/4/045010

Cited by 19 publications

(70 citation statements)

References 42 publications

(142 reference statements)

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“…For instance, Sylvester and Uhlmann treated in [9,18] the uniqueness of solution; Curtis, Ingerman and Morrow have worked on critical circular planar networks conductivity reconstruction [12,13,14,16]; Borcea, Druskin, Guevara and Mamonov have gone into EIT problems in depth and their last works on the subject treat numerical conductivity reconstruction [6,7,8].…”

Section: Introductionmentioning

confidence: 99%

Overdetermined partial boundary value problems on finite networks

Araúz

Carmona

Encinas

2015

Journal of Mathematical Analysis and Applications

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In this work we define a class of non self-adjoint boundary value problems on finite networks associated with Schrödinger operators. The novelty lies on the fact that on a part of the boundary no data is prescribed, whereas in another part of the boundary both the values of the function as of its normal derivative are given. We show that overdetermined partial boundary value problems are the key for solving inverse boundary value problems on finite networks, since they provide the theoretical foundations of the recovery algorithm. We analyze the uniqueness and existence of solution of overdetermined partial boundary value problems through the non-singularity of the partial Dirichlet-to-Neumann maps. These maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed. Afterwards, we execute a full conductance recovery for spider networks.

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Section: Introductionmentioning

confidence: 99%

Overdetermined partial boundary value problems on finite networks

Araúz

Carmona

Encinas

2015

Journal of Mathematical Analysis and Applications

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“…The results in [11] show a trade-off between having undistorted images and resolution distributed throughout Ω. To eliminate distortions, the transformed conductivity should remain isotropic, which means that the coordinate transformation must be conformal.…”

Section: Motivation and Outline Of The Results In This Papermentioning

confidence: 98%

“…It is because of it and the essential one dimensional nature of the grids, that we get the trade-off studied in [11]. Our main result in this paper is the introduction of truly two dimensional optimal grids, with pyramidal topology, that is naturally suited for the partial measurements setup.…”

Section: Motivation and Outline Of The Results In This Papermentioning

confidence: 99%

“…This is done in [11] with conformal or extremal quasiconformal mappings that take the boundary points in B A to equidistantly distributed points on B. Then, the grids for the transformed problem are computed with the approach in [8,27].…”

Section: Motivation and Outline Of The Results In This Papermentioning

confidence: 99%

“…Then, we show how to use the networks to define the optimal grids and to approximate the unknown conductivity. We assess the performance of our approach with numerical simulations and compare the results with those in [11]. …”

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confidence: 99%

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Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

et al. 2010

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Abstract. We introduce an inversion algorithm for electrical impedance tomography (EIT) with partial boundary measurements, in two dimensions. It gives stable and fast reconstructions using sparse parameterizations of the unknown conductivity on optimal grids that are computed as part of the inversion. We follow the approach in [8,27] that connects inverse discrete problems for resistor networks to continuum EIT problems, using optimal grids. The algorithm in [8,27] is based on circular resistor networks, and solves the EIT problem with full boundary measurements. It is extended in [11] to EIT with partial boundary measurements, using extremal quasiconformal mappings that transform the problem to one with full boundary measurements. Here we introduce a different class of optimal grids, based on resistor networks with pyramidal topology, that is better suited for the partial measurements setup. We prove the unique solvability of the discrete inverse problem for these networks, and develop an algorithm for finding them from the measurements of the DtN map. Then, we show how to use the networks to define the optimal grids and to approximate the unknown conductivity. We assess the performance of our approach with numerical simulations and compare the results with those in [11].

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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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Circular resistor networks for electrical impedance tomography with partial boundary measurements

Cited by 19 publications

References 42 publications

Overdetermined partial boundary value problems on finite networks

Overdetermined partial boundary value problems on finite networks

Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

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

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