Electrical impedance tomography: 3D reconstructions using scattering transforms

Delbary, Fabrice; Hansen, Per Christian; Knudsen, Kim

doi:10.1080/00036811.2011.598863

Cited by 22 publications

(26 citation statements)

References 30 publications

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“…where a, ξ ∈ ℝ 2 with ξ • a = 0 and |ξ| = |a|, to Laplace's equation for the linearized inversion method. Three-dimensional EIT algorithms based on CGO solutions were discussed and implemented in [110,111,112,113,114,115]. In the D-bar method presented here, the CGO solutions are special solutions to the Schrödinger equation involving an artificial complex frequency variable k, and they are the key to the direct computation of the conductivity.…”

Section: D-barmentioning

confidence: 99%

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

Martins

Sato

Moura

et al. 2019

Annual Reviews in Control

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Electrical Impedance Tomography (EIT) is under fast development, the present paper is a review of some procedures that are contributing to improve spatial resolution and material properties accuracy, admitivitty or impeditivity accuracy. A review of EIT medical applications is presented and they were classified into three broad categories: ARDS patients, obstructive lung diseases and perioperative patients. The use of absolute EIT image may enable the assessment of absolute lung volume, which may significantly improve the clinical acceptance of EIT. The Control Theory, the State Observers more specifically, have a developed theory that can be used for the design and operation of EIT devices. Electrode placement, current injection strategy and electrode electric potential measurements strategy should maximize the number of observable and controllable directions of the state vector space. A non-linear stochastic state observer, the Unscented Kalman Filter, is used directly for the reconstruction of absolute EIT images. Historically, difference images were explored first since they are more stable in the presence of modelling errors. Absolute images require more detailed models of contact impedance, stray capacitance and properly refined finite element mesh where the electric potential gradient is high. Parallelization of the forward *

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Section: D-barmentioning

confidence: 99%

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

Martins

Sato

Moura

et al. 2019

Annual Reviews in Control

View full text Add to dashboard Cite

show abstract

“…For example, the complex geometrical optics solutions to the conductivity equation can be used to reconstruct the conductivity distributions [23]. However, the electrode configuration in [23] is not suitable to monitor the flows in the pipe, since electrodes point electrodes located at the Gaussian quadrature points on the sphere. The conductivity at the boundary can be recovered with reasonable accuracy using practically realizable measurements [24].…”

Section: Introductionmentioning

confidence: 99%

Direct Image Reconstruction for 3-D Electrical Resistance Tomography by Using the Factorization Method and Electrodes on a Single Plane

Cao

2013

IEEE Trans. Instrum. Meas.

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A direct reconstruction method for three-dimensional (3-D) electrical resistance tomography was introduced by using the factorization method. Compared with the traditional image reconstruction algorithms based on the sensitivity/Jacobian matrix, the conductivity distribution in any part of the 3-D region of interest can be obtained directly and independently. A new way to calculate the Neumann-to-Dirichlet map was also introduced by using the adjacent current pattern. Several phantoms were constructed for image reconstruction in three dimensions. The data were collected from 16 electrodes on a single cross section, which can be only used to produce two-dimensional images in the literature. Neither matrix inversion nor iteration was used in the process of image reconstruction. The reconstructed results validated the feasibility of the method.

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“…In a forthcoming article the authors intend to apply the Green's function constructed here to the problem of reconstruction. One expects that in the context of computational algorithms this Green's function would open the door to direct inversion methods for partial data Calderón problems in n ≥ 3 which is parallel to the full data case examined in [1,[10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Partial data inverse problem with 𝐿^{𝑛/2} potentials

Chung

Tzou

2020

Trans. Amer. Math. Soc. Ser. B

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We construct an explicit Green’s function for the conjugated Laplacian e − ω ⋅ x / h Δ e − ω ⋅ x / h e^{-\omega \cdot x/h}\Delta e^{-\omega \cdot x/h} , which lets us control our solutions on roughly half of the boundary. We apply the Green’s function to solve a partial data inverse problem for the Schrödinger equation with potential q ∈ L n / 2 q \in L^{n/2} . Separately, we also use this Green’s function to derive L p L^p Carleman estimates similar to the ones in Kenig-Ruiz-Sogge [Duke Math. J. 55 (1987), pp. 329–347], but for functions with support up to part of the boundary. Unlike many previous results, we did not obtain the partial data result from the boundary Carleman estimate—rather, both results stem from the same explicit construction of the Green’s function. This explicit Green’s function has potential future applications in obtaining direct numerical reconstruction algorithms for partial data Calderón problems which is presently only accessible with full data [Inverse Problems 27 (2011)].

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Electrical impedance tomography: 3D reconstructions using scattering transforms

Cited by 22 publications

References 30 publications

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

Direct Image Reconstruction for 3-D Electrical Resistance Tomography by Using the Factorization Method and Electrodes on a Single Plane

Partial data inverse problem with 𝐿^{𝑛/2} potentials

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