## Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

Citations: | 4 - 4 self |

### BibTeX

@MISC{Borcea_pyramidalresistor,

author = {L. Borcea and V. Druskin and A. V. Mamonov and F. Guevara Vasquez},

title = {Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements},

year = {}

}

### OpenURL

### Abstract

### Citations

638 | Partial differential equations
- Evans
- 1998
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Citation Context ... (4.2) where y ∈ Ω is the source location. Our numerical simulations are for domain Ω = D the unit disk, and Ω = R 2 − the lower half plane, respectively. In both cases we can write G(x,y) explicitly =-=[26]-=- when σ ≡ 1, as needed in the computation of the optimal grids. For Ω = D we have [26] GD(x,y) = 1 (−log |x − y| + log |y|(|x − ˜y|)), (4.3) 2π where ˜y = y/|y| 2 and | · | is the Euclidean norm. When... |

538 |
Direct Methods for Sparse Matrices
- DUFF, ERISMAN, et al.
- 1986
(Show Context)
Citation Context ....12) .. Um−2 ⎦ I 0 Dm−1 0 I Lm−2 where all blocks Dj, j = 1,...,m − 1, are non-singular, because KII is invertible. If we denote the diagonal blocks of K −1 II by Zj, j = 1,...,m − 1, it can be shown =-=[23, 25, 44]-=- that they satisfy Zm−1 = D −1 m−1 , (B.13) Zj = D −1 j + UjZj+1Lj, j = m − 2,...,1. (B.14) Of particular interest to us is (B.13), which gives Zm−1 = D −1 m−1 = ( K −1) II , hence ( ) SS −1 K II SS i... |

420 |
Regularization of Inverse Problems
- Engl, Hank, et al.
- 1996
(Show Context)
Citation Context ...e measurements are made. This means that if we use inadequate parameterizations of the unknown σ, on grids that are too fine inside Ω, the numerical inversion will be unstable and must be regularized =-=[24]-=-. The question is then how to find proper parameterizations of σ, on grids that capture correctly the resolution limits, to get stable images without additional regularization that typically requires ... |

116 |
Global uniqueness for a two-dimensional inverse boundary value problem
- Nachman
- 1996
(Show Context)
Citation Context ... ∂u (x), x ∈ B, (1.3) ∂ν and ν is the outer unit normal at B. EIT with full boundary measurements refers to the ideal case with complete knowledge of the DtN map. It is uniquely solvable as proved in =-=[40, 12]-=- under some regularity assumptions on σ, and in [3] for bounded σ. We consider the EIT problem with partial boundary measurements on the accessible subset BA of B. The inaccessible boundary BI = B \ B... |

86 |
On an inverse boundary value problem. Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro
- Calderón
- 1980
(Show Context)
Citation Context ... Introduction. We present a novel approach for the numerical approximation of solutions of electrical impedance tomography (EIT) with partial boundary measurements, in two dimensions. The EIT problem =-=[13, 6]-=- is to find the conductivity σ(x) in a simply connected domain Ω ⊂ R 2 , given simultaneous measurements of currents and voltages at the boundary B of Ω. More explicitly, σ(x) is the coefficient in th... |

66 |
Stable determination of conductivity by boundary measurements, Applicable Anal
- Alessandrini
- 1988
(Show Context)
Citation Context ... case of complete knowledge of the DtN map. By exponential instability we mean that the sup norm of perturbations of σ is bounded in terms of the logarithm of the operator norm of perturbations of Λσ =-=[1, 5, 33]-=-. The bounds are sharp [38], but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω. The trade-off between stability and resolution is studied in [2, 32,... |

60 |
Calderón’s inverse conductivity problem in the plane
- Astala, Päivärinta
- 2006
(Show Context)
Citation Context ...at B. EIT with full boundary measurements refers to the ideal case with complete knowledge of the DtN map. It is uniquely solvable as proved in [40, 12] under some regularity assumptions on σ, and in =-=[3]-=- for bounded σ. We consider the EIT problem with partial boundary measurements on the accessible subset BA of B. The inaccessible boundary BI = B \ BA { is assumed grounded. That σ is determined uniqu... |

56 |
Determining the conductivity by boundary measurements
- Kohn, Vogelius
- 1984
(Show Context)
Citation Context ...nts on the accessible subset BA of B. The inaccessible boundary BI = B \ BA { is assumed grounded. That σ is determined uniquely by the set of Cauchy data u| , σ BA ∂u ∣ } , when u| = 0, follows from =-=[19, 20, 34, 35]-=- for BI ∣ ∂ν BA real-analytic or piecewise real-analytic σ, and from [28] for σ ∈ C 3+ɛ ( ¯ Ω), with ɛ > 0. Because we are concerned with numerical inversion, we work with finitely many measurements‡ ... |

50 |
Existence and uniqueness for a electrode models for electric current computed tomography
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- 1992
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Citation Context ...setups. The partial data case corresponds to suppχj ⊂ BA for (2.17), or xj ∈ BA for (2.20). Other measurement operators that use more accurate electrode models, such as the “complete electrode” model =-=[42]-=- can be used in principle. The crucial question is whether the range of the operators belongs to the set Dn of DtN maps of well connected networks. This is the case for the operators (2.20) and (2.17)... |

44 | Electrical Impedance Tomography
- Borcea
- 2002
(Show Context)
Citation Context ... Introduction. We present a novel approach for the numerical approximation of solutions of electrical impedance tomography (EIT) with partial boundary measurements, in two dimensions. The EIT problem =-=[13, 6]-=- is to find the conductivity σ(x) in a simply connected domain Ω ⊂ R 2 , given simultaneous measurements of currents and voltages at the boundary B of Ω. More explicitly, σ(x) is the coefficient in th... |

41 |
An anisotropic inverse boundary value problem
- Sylvester
- 1990
(Show Context)
Citation Context ... exactly the same. Moreover, formula (4.11) is valid for any region conformally equivalent to the unit disk. This follows from the invariance of the DtN map under conformal coordinate transformations =-=[43]-=-. The behavior of the kernel Kσ(x,y) for general σ is similar to (4.11) in the sense that away from the diagonal x = y, it admits the representation [31] Kσ(x,y) = − k(x,y) . (4.12) π|x − y| 2 Here k(... |

37 | Exponential instability in an inverse problem for the Schrodinger equation
- Mandache
- 2013
(Show Context)
Citation Context ...e DtN map. By exponential instability we mean that the sup norm of perturbations of σ is bounded in terms of the logarithm of the operator norm of perturbations of Λσ [1, 5, 33]. The bounds are sharp =-=[38]-=-, but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω. The trade-off between stability and resolution is studied in [2, 32, 41] for linearized, full b... |

34 | Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions - Brown, Uhlmann - 1997 |

33 |
Morrow: Circular planar graphs and resistor networks, Linear Algebra and its Applications 283
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Citation Context ...lem for networks. The question of whether the discrete inverse problem is uniquely solvable is closely related to the topology of the graph Γ. For circular planar graphs, the question was resolved in =-=[15]-=-, using the theory of critical networks. Let Γ ′ be the graph obtained by removing one edge in Γ = (Y,E). The edge can be removed by deletion or by contraction. Then, the network with graph Γ is calle... |

28 |
Distinguishability of conductivities by electric current computed tomography
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(Show Context)
Citation Context ... 5, 33]. The bounds are sharp [38], but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω. The trade-off between stability and resolution is studied in =-=[2, 32, 41]-=- for linearized, full boundary data EIT. The results in [2] give explicit reconstructions of small perturbations δσ of a constant conductivity, which are then used to assess the stability and resoluti... |

27 |
On computing certain elements of the inverse of a sparse matrix
- Erisman, Tinney
- 1975
(Show Context)
Citation Context ....12) .. Um−2 ⎦ I 0 Dm−1 0 I Lm−2 where all blocks Dj, j = 1,...,m − 1, are non-singular, because KII is invertible. If we denote the diagonal blocks of K −1 II by Zj, j = 1,...,m − 1, it can be shown =-=[23, 25, 44]-=- that they satisfy Zm−1 = D −1 m−1 , (B.13) Zj = D −1 j + UjZj+1Lj, j = m − 2,...,1. (B.14) Of particular interest to us is (B.13), which gives Zm−1 = D −1 m−1 = ( K −1) II , hence ( ) SS −1 K II SS i... |

25 |
Determining the conductivity by boundary measurements II, interior results
- Kohn, Vogelius
- 1985
(Show Context)
Citation Context ...nts on the accessible subset BA of B. The inaccessible boundary BI = B \ BA { is assumed grounded. That σ is determined uniquely by the set of Cauchy data u| , σ BA ∂u ∣ } , when u| = 0, follows from =-=[19, 20, 34, 35]-=- for BI ∣ ∂ν BA real-analytic or piecewise real-analytic σ, and from [28] for σ ∈ C 3+ɛ ( ¯ Ω), with ɛ > 0. Because we are concerned with numerical inversion, we work with finitely many measurements‡ ... |

23 |
Formation of a sparse bus impedance matrix and its application to short circuit study
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- 1973
(Show Context)
Citation Context ....12) .. Um−2 ⎦ I 0 Dm−1 0 I Lm−2 where all blocks Dj, j = 1,...,m − 1, are non-singular, because KII is invertible. If we denote the diagonal blocks of K −1 II by Zj, j = 1,...,m − 1, it can be shown =-=[23, 25, 44]-=- that they satisfy Zm−1 = D −1 m−1 , (B.13) Zj = D −1 j + UjZj+1Lj, j = m − 2,...,1. (B.14) Of particular interest to us is (B.13), which gives Zm−1 = D −1 m−1 = ( K −1) II , hence ( ) SS −1 K II SS i... |

21 | Réseaux électriques planaires
- VERDIÈRE
- 1994
(Show Context)
Citation Context ...ional EIT was proposed and analyzed in [8, 27], for the case of full boundary measurements. It is based on the rigorous theory of discrete inverse problems for circular resistor networks developed in =-=[14, 15, 29, 17, 18]-=-. These networks arise in five point stencil finite volumes discretizations of (1.1)–(1.2), on the optimal grids. The networks are critical, which means that they have no redundant connections and are... |

16 |
The unique solution of the inverse problem in electrical surveying and electrical well logging for piecewise-constant conductivity
- Druskin
- 1982
(Show Context)
Citation Context ...nts on the accessible subset BA of B. The inaccessible boundary BI = B \ BA { is assumed grounded. That σ is determined uniquely by the set of Cauchy data u| , σ BA ∂u ∣ } , when u| = 0, follows from =-=[19, 20, 34, 35]-=- for BI ∣ ∂ν BA real-analytic or piecewise real-analytic σ, and from [28] for σ ∈ C 3+ɛ ( ¯ Ω), with ɛ > 0. Because we are concerned with numerical inversion, we work with finitely many measurements‡ ... |

15 |
On a characterization of the kernel of the Dirichlet-toNeumann map for a planar region
- Ingerman, Morrow
- 1998
(Show Context)
Citation Context ...the first order pseudodifferential operator Λσ can be written in an integral form as ∫ (Λσφ)(x) = Kσ(x,y)φ(y)dSy, x ∈ B, (2.19) B where Kσ(x,y) is a symmetric kernel continuous away from the diagonal =-=[31]-=-. The pointwise measurement operator Mn is defined at the points xj ∈ B, j = 1,...,n, by { Kσ(xi,xj), i ̸= j, (Mn(Λσ))i,j = − ∑ Kσ(xi,xk), i = j. (2.20) k̸=i These definitions do not distinguish betwe... |

14 |
Stability of the inverse conductivity problem in the plane for less regular conductivities
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Citation Context ... case of complete knowledge of the DtN map. By exponential instability we mean that the sup norm of perturbations of σ is bounded in terms of the logarithm of the operator norm of perturbations of Λσ =-=[1, 5, 33]-=-. The bounds are sharp [38], but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω. The trade-off between stability and resolution is studied in [2, 32,... |

14 |
Gaussian spectral rules for the three-point second differences: I. A two-point positive definite problem in a semi-infinite domain
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Citation Context ...ements. 3 accessible boundary, where we make the measurements, and are coarse away from it, thus capturing the expected loss of resolution of the images. Optimal grids were introduced and analyzed in =-=[4, 21, 22, 30]-=- for forward problems. Then, they were used in [7] for Sturm-Liouville inverse spectral problems in one dimension. The main result there is that parameterizations on optimal grids are necessary and su... |

13 |
Optimal finite difference grids for direct and inverse SturmLiouville problems, Inverse Problems
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Citation Context ... and are coarse away from it, thus capturing the expected loss of resolution of the images. Optimal grids were introduced and analyzed in [4, 21, 22, 30] for forward problems. Then, they were used in =-=[7]-=- for Sturm-Liouville inverse spectral problems in one dimension. The main result there is that parameterizations on optimal grids are necessary and sufficient for convergence of solutions of discrete ... |

13 |
Finding the conductors in circular networks from boundary measurements
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Citation Context ...ional EIT was proposed and analyzed in [8, 27], for the case of full boundary measurements. It is based on the rigorous theory of discrete inverse problems for circular resistor networks developed in =-=[14, 15, 29, 17, 18]-=-. These networks arise in five point stencil finite volumes discretizations of (1.1)–(1.2), on the optimal grids. The networks are critical, which means that they have no redundant connections and are... |

13 | Morrow: Inverse Problems for Electrical Networks - Curtis, A - 2000 |

12 | Global uniqueness from partial Cauchy data in two dimensions, Arxiv preprint arXiv:0810.2286 - Imanuvilov, Uhlmann, et al. - 2008 |

11 |
Stability and resolution analysis of a linearized problem in electrical impedance tomography
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(Show Context)
Citation Context ... 5, 33]. The bounds are sharp [38], but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω. The trade-off between stability and resolution is studied in =-=[2, 32, 41]-=- for linearized, full boundary data EIT. The results in [2] give explicit reconstructions of small perturbations δσ of a constant conductivity, which are then used to assess the stability and resoluti... |

11 |
Application of the difference Gaussian rules to solution of hyperbolic problems
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Citation Context ...ements. 3 accessible boundary, where we make the measurements, and are coarse away from it, thus capturing the expected loss of resolution of the images. Optimal grids were introduced and analyzed in =-=[4, 21, 22, 30]-=- for forward problems. Then, they were used in [7] for Sturm-Liouville inverse spectral problems in one dimension. The main result there is that parameterizations on optimal grids are necessary and su... |

11 |
Gaussian spectral rules for second order finite-difference schemes, Numerical Algorithms
- Druskin, Knizhnerman
(Show Context)
Citation Context ...ements. 3 accessible boundary, where we make the measurements, and are coarse away from it, thus capturing the expected loss of resolution of the images. Optimal grids were introduced and analyzed in =-=[4, 21, 22, 30]-=- for forward problems. Then, they were used in [7] for Sturm-Liouville inverse spectral problems in one dimension. The main result there is that parameterizations on optimal grids are necessary and su... |

11 | Discrete and continuous Dirichlet-to-Neumann maps in the layered case
- Ingerman
(Show Context)
Citation Context ...ional EIT was proposed and analyzed in [8, 27], for the case of full boundary measurements. It is based on the rigorous theory of discrete inverse problems for circular resistor networks developed in =-=[14, 15, 29, 17, 18]-=-. These networks arise in five point stencil finite volumes discretizations of (1.1)–(1.2), on the optimal grids. The networks are critical, which means that they have no redundant connections and are... |

11 |
A direct reconstruction algorithm for electrical impedance tomography
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(Show Context)
Citation Context ...r the reconstructions in the unit disk we use the same test conductivity functions as in [8, 27, 11]. The first one is a smooth function sigX, the other is a piecewise constant chest phantom phantom1 =-=[39]-=-. Both conductivities are shown in figure 6. The high contrast conductivity used in section 5.3.3 is simply { [ ] π 3π 1, θ ∈ σ(r,θ) = 2 , 2 , C0, θ ∈ ( 0, π ) ( 3π 2 ∪ 2 ,2π) (5.2) , where C0 is the ... |

10 |
On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids
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Citation Context ...ft: n = 6; right: n = 7. Boundary nodes vj, j = 1,...,n are ×, interior nodes are ◦. σ ≡ 1. Illustrations of the good extrapolation properties of optimal grids, for various measurement setups, are in =-=[9, 8, 27, 7, 11]-=-. There is only one part of the inversion algorithm outlined above that is sensitive to the measurement setup. It is the definition of the optimal grid, and therefore of the reconstruction mapping Qn.... |

10 |
Optimal finite difference grids and rational approximations of the square root. I. Elliptic problems
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(Show Context)
Citation Context |

9 |
Electrical impedance tomography with resistor networks
- Borcea, Druskin, et al.
(Show Context)
Citation Context ... 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 probl... |

9 |
On uniqueness of the determination of the three-dimensional underground structures from surface measurements with variously positioned steady-state or monochromatic field sources”, Sov. Phys.–Solid Earth 21
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Citation Context |

7 | On the Parametrization of Ill-posed Inverse Problems Arising from Elliptic Partial Differential Equations
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Citation Context ... 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 probl... |

7 |
First-order system least squares and electrical impedance tomography
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(Show Context)
Citation Context ...ture correctly the resolution limits, to get stable images without additional regularization that typically requires prior information about σ. The distinguishability grids proposed in [32] (see also =-=[37, 36]-=-) capture qualitatively the loss of resolution, but they are defined with a linearization approach whose accuracy is not understood. Here we follow the ideas in [8, 27, 11] and parametrize σ on optima... |

4 |
Reseaux electriques planaires II
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Citation Context |

4 |
Theoretical limits to sensitivity and resolution in impedance imaging
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Citation Context ... 5, 33]. The bounds are sharp [38], but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω. The trade-off between stability and resolution is studied in =-=[2, 32, 41]-=- for linearized, full boundary data EIT. The results in [2] give explicit reconstructions of small perturbations δσ of a constant conductivity, which are then used to assess the stability and resoluti... |

3 |
Regularized d-bar method for the inverse conductivity problem
- Knudsen, Lassas, et al.
(Show Context)
Citation Context ... case of complete knowledge of the DtN map. By exponential instability we mean that the sup norm of perturbations of σ is bounded in terms of the logarithm of the operator norm of perturbations of Λσ =-=[1, 5, 33]-=-. The bounds are sharp [38], but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω. The trade-off between stability and resolution is studied in [2, 32,... |

2 |
Solving the discrete EIT problem with optimization techniques, 2007. Schlumberger-Doll Report
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(Show Context)
Citation Context ... optimization methods for network recovery, because we cannot speak of regularity assumptions as is done, for example, in total variation regularization approaches for continuum EIT. We have tried in =-=[10]-=- a Gauss-Newton iterative optimization method regularized with an SVD truncation of the Jacobian. All the results in this paper are with layer peeling, which we regularize by restricting n, i.e. the s... |