## Circular resistor networks for electrical impedance tomography with partial boundary measurements.

Citations: | 5 - 4 self |

### BibTeX

@MISC{Borcea_circularresistor,

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

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

year = {}

}

### OpenURL

### Abstract

Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on so-called optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting trade-off between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary. 1.

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Citation Context ... convex duality.Circular resistor networks for the EIT with partial measurements. 3 parametrization. If we seek too many parameters, the numerical method becomes unstable, and it must be regularized =-=[25]-=- using for example prior assumptions on σ. We assume no such prior information, and use instead a regularization approach based on sparse parametrizations of the conductivity. Regularization by sparse... |

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Citation Context ... Λ A ( ) ∂u σ φ|BA = σ ∣ , (1.4) ∂ν ∣ BA where φ from (1.2) obeys an additional condition supp φ ⊂ BA. The uniqueness of solution of the EIT problem with full boundary measurements was established in =-=[48, 42]-=- under some smoothness assumptions on σ, and more recently in [2], for bounded σ. The uniqueness of solution of the EIT problem with partial boundary measurements, and for real-analytic or piecewise r... |

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Citation Context ...ion This paper is concerned with the numerical approximation of solutions of the inverse problem of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. In EIT =-=[13, 6]-=- we wish to determine the positive and bounded coefficient σ in the elliptic equation ∇ · (σ(x)∇u(x)) = 0, x ∈ Ω, (1.1) from measurements of the Dirichlet-to-Neumann (DtN) map Λσ. The domain Ω is an o... |

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Citation Context ...t to our formulation is given in [27], for σ of class C3+ɛ in the closure of the domain, where ɛ > 0. Even with full boundary measurements the EIT problem is unstable. There exist stability estimates =-=[1, 4]-=- under some assumptions on the regularity of σ, but they cannot be better than logarithmic [40]. That is, given two sufficiently regular conductivities σ1 and σ2, the following estimate holds ∣ ∣ , (1... |

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Citation Context ...dditional condition supp φ ⊂ BA. The uniqueness of solution of the EIT problem with full boundary measurements was established in [48, 42] under some smoothness assumptions on σ, and more recently in =-=[2]-=-, for bounded σ. The uniqueness of solution of the EIT problem with partial boundary measurements, and for real-analytic or piecewise real-analytic σ, follows from [21, 22, 33, 34]. The first global u... |

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Citation Context ...tions on σ, and more recently in [2], for bounded σ. The uniqueness of solution of the EIT problem with partial boundary measurements, and for real-analytic or piecewise real-analytic σ, follows from =-=[21, 22, 33, 34]-=-. The first global uniqueness result was obtained more recently in [12], in three or more dimensions and for a restrictive measurement setup. The uniqueness result that is relevant to our formulation ... |

44 | Electrical Impedance Tomography
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Citation Context ...ion This paper is concerned with the numerical approximation of solutions of the inverse problem of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. In EIT =-=[13, 6]-=- we wish to determine the positive and bounded coefficient σ in the elliptic equation ∇ · (σ(x)∇u(x)) = 0, x ∈ Ω, (1.1) from measurements of the Dirichlet-to-Neumann (DtN) map Λσ. The domain Ω is an o... |

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Citation Context ...lem with measurements of Λ˜σ on B. Here ˜σ is the transformed conductivity, and it is matrix valued (anisotropic), in general. Anisotropic conductivities are not uniquely recoverable from the DtN map =-=[47]-=-, so our mappings should either preserve the isotropy of σ (i.e., be conformal) or, at least minimize its anisotropy (i.e., be extremal quasiconformal) [35]. In other words, we can obtain grids for pa... |

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Citation Context ...> 0. Even with full boundary measurements the EIT problem is unstable. There exist stability estimates [1, 4] under some assumptions on the regularity of σ, but they cannot be better than logarithmic =-=[40]-=-. That is, given two sufficiently regular conductivities σ1 and σ2, the following estimate holds ∣ ∣ , (1.5) ‖σ1 − σ2‖L∞ (D) ≤ C ∣log ‖Λσ1 − Λσ2‖H1/2 (B)→H−1/2(B) ∣ −α where C and α are positive const... |

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Citation Context ...of a circular resistor network with critical graph. This is shown in [11, 49] using the complete characterization of the set Dn of DtN maps of circular resistor networks with critical graphs given in =-=[15, 14, 18]-=- and the characterization of the kernel of the continuum DtN map in [30]. Thus, the measurements Mn(Λσ) are consistent with a unique resistor network C((n−1)/2,n), with odd number n of boundary nodes ... |

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Citation Context ...e we make the measurements, and they are coarse inside the domain, thus capturing the gradual loss of resolution of the reconstructions away from B. Other adaptive grids for EIT have been proposed in =-=[31, 38, 37]-=-. They are called distinguishability grids because they are constructed with a linearization argument that looks for the smallest support of a perturbation δσ at a given location in the domain, that c... |

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Citation Context ...tions on σ, and more recently in [2], for bounded σ. The uniqueness of solution of the EIT problem with partial boundary measurements, and for real-analytic or piecewise real-analytic σ, follows from =-=[21, 22, 33, 34]-=-. The first global uniqueness result was obtained more recently in [12], in three or more dimensions and for a restrictive measurement setup. The uniqueness result that is relevant to our formulation ... |

21 | Réseaux électriques planaires
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Citation Context ... two dimensional EIT with full boundary measurements was proposed and analyzed in [11, 49]. It is based on the rigorous theory of discrete inverse problems for circular resistor networks developed in =-=[14, 15, 28, 17, 18]-=-. The circular networks arise in the discretization of equation (1.1) with a five point stencil finite volumes scheme on the optimal grids computed as part of the inverse problem. The networks are cri... |

20 |
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Citation Context ...d Ψ in its decomposition map the unit disk conformally onto polygons comprised of a number of rectangular strips. But conformal mappings of the unit disk to polygons are Schwartz-Christoffel mappings =-=[20]-=-, given by the general formula ∫ z ∏N S(z) = A + B q=1 ( 1 − ζ zq ) αq−1 dζ, (3.23) where A,B ∈ C are constants, N is the number of vertices of a polygon, zq ∈ B are the pre-images of the vertices and... |

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Citation Context ...tions on σ, and more recently in [2], for bounded σ. The uniqueness of solution of the EIT problem with partial boundary measurements, and for real-analytic or piecewise real-analytic σ, follows from =-=[21, 22, 33, 34]-=-. The first global uniqueness result was obtained more recently in [12], in three or more dimensions and for a restrictive measurement setup. The uniqueness result that is relevant to our formulation ... |

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Citation Context ...ing the complete characterization of the set Dn of DtN maps of circular resistor networks with critical graphs given in [15, 14, 18] and the characterization of the kernel of the continuum DtN map in =-=[30]-=-. Thus, the measurements Mn(Λσ) are consistent with a unique resistor network C((n−1)/2,n), with odd number n of boundary nodes and conductance γ. This is not sufficient however to obtain an approxima... |

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Citation Context ...t to our formulation is given in [27], for σ of class C3+ɛ in the closure of the domain, where ɛ > 0. Even with full boundary measurements the EIT problem is unstable. There exist stability estimates =-=[1, 4]-=- under some assumptions on the regularity of σ, but they cannot be better than logarithmic [40]. That is, given two sufficiently regular conductivities σ1 and σ2, the following estimate holds ∣ ∣ , (1... |

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Citation Context ...ased on rational approximation techniques. They are called optimal because they give spectral accuracy of the DtN map with finite volumes on coarse grids. Optimal grids were introduced and anlyzed in =-=[3, 23, 24, 29]-=-, for forward problems. The first inversion method on optimal grids was proposed in [7], for SturmLiouville inverse spectral problems in one dimension. Then, it was shown in [8] that optimal grids pro... |

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Citation Context ...acy of the DtN map with finite volumes on coarse grids. Optimal grids were introduced and anlyzed in [3, 23, 24, 29], for forward problems. The first inversion method on optimal grids was proposed in =-=[7]-=-, for SturmLiouville inverse spectral problems in one dimension. Then, it was shown in [8] that optimal grids provide a necessary and sufficient condition for convergence of solutions of discrete inve... |

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Citation Context ... two dimensional EIT with full boundary measurements was proposed and analyzed in [11, 49]. It is based on the rigorous theory of discrete inverse problems for circular resistor networks developed in =-=[14, 15, 28, 17, 18]-=-. The circular networks arise in the discretization of equation (1.1) with a five point stencil finite volumes scheme on the optimal grids computed as part of the inverse problem. The networks are cri... |

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Citation Context ...t B. EIT with full boundary measurements corresponds to the case where all possible boundary excitations and measurements are available. We consider the EIT problem with partial boundary measurements =-=[27]-=- on the accessible subset BA of the boundary. The inaccessible boundary BI = B \ BA is assumed grounded‡. The problem is to determine σ given the knowledge of the operator ΛA σ : H1/2 (BA) → H−1/2 (BA... |

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Citation Context ...ased on rational approximation techniques. They are called optimal because they give spectral accuracy of the DtN map with finite volumes on coarse grids. Optimal grids were introduced and anlyzed in =-=[3, 23, 24, 29]-=-, for forward problems. The first inversion method on optimal grids was proposed in [7], for SturmLiouville inverse spectral problems in one dimension. Then, it was shown in [8] that optimal grids pro... |

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Citation Context ...ased on rational approximation techniques. They are called optimal because they give spectral accuracy of the DtN map with finite volumes on coarse grids. Optimal grids were introduced and anlyzed in =-=[3, 23, 24, 29]-=-, for forward problems. The first inversion method on optimal grids was proposed in [7], for SturmLiouville inverse spectral problems in one dimension. Then, it was shown in [8] that optimal grids pro... |

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Citation Context ...ductivity. We are not aware of any stability results for EIT with partial boundary measurements (1.4). There is a stability estimate in dimensions three or higher for a less general measurement setup =-=[26]-=-, which is of log-log type. Nevertheless, it is clear that if stability estimates existed, they could not be better than the estimate (1.5) for the full boundary measurements case. Naturally, any nume... |

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Citation Context ... two dimensional EIT with full boundary measurements was proposed and analyzed in [11, 49]. It is based on the rigorous theory of discrete inverse problems for circular resistor networks developed in =-=[14, 15, 28, 17, 18]-=-. The circular networks arise in the discretization of equation (1.1) with a five point stencil finite volumes scheme on the optimal grids computed as part of the inverse problem. The networks are cri... |

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Citation Context ... They are shown in figure 8. The first one is a smooth conductivity (sigX), given by the superposition of two Gaussians. The second one is piecewise constant (phantom1), and it models a chest phantom =-=[41]-=-. It appears from the examples of optimal grids in figures 4 and 7, that the reconstructions will have better resolution near the accessible boundary. To explore this phenomenon, we rotate the accessi... |

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Citation Context ...anlyzed in [3, 23, 24, 29], for forward problems. The first inversion method on optimal grids was proposed in [7], for SturmLiouville inverse spectral problems in one dimension. Then, it was shown in =-=[8]-=- that optimal grids provide a necessary and sufficient condition for convergence of solutions of discrete inverse spectral problems to the true solution of the continuum problem. The first inversion m... |

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Electrical impedance tomography with resistor networks
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Citation Context ...tract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in =-=[11, 49]-=- for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on so-called opt... |

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First-order system least squares and electrical impedance tomography
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Citation Context ...e we make the measurements, and they are coarse inside the domain, thus capturing the gradual loss of resolution of the reconstructions away from B. Other adaptive grids for EIT have been proposed in =-=[31, 38, 37]-=-. They are called distinguishability grids because they are constructed with a linearization argument that looks for the smallest support of a perturbation δσ at a given location in the domain, that c... |

7 | On the Parametrization of Ill-posed Inverse Problems Arising from Elliptic Partial Differential Equations
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Citation Context ...tract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in =-=[11, 49]-=- for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on so-called opt... |

5 |
The inverse conductivity problem with an imperfectly known boundary
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Citation Context ...e not uniquely recoverable from the DtN map [47], so our mappings should either preserve the isotropy of σ (i.e., be conformal) or, at least minimize its anisotropy (i.e., be extremal quasiconformal) =-=[35]-=-. In other words, we can obtain grids for partial boundary measurements that have good approximation properties for a class of conductivity functions, including constant ones, via extremal quasiconfor... |

5 |
On the existence of extremal Teichmüller mappings
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Citation Context ...approximation properties in some class of conductivity functions. Our main result in this paper is that the grids can be constructed with an elegant approach based on extremal quasiconformal mappings =-=[45]-=- that transform the problem with partial measurements to a problem with full boundary measurements. There are many transformations (diffeomorphisms) of the unit disk D to itself that take the EIT prob... |

4 | Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems
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Citation Context ...otropy. This is in fact a consequence of the topology of the grids and resistor networks, and it may be circumvented by considering different topologies, as shown in [39] and in a forthcoming article =-=[10]-=-. We end with the observation that since we proved that the optimal grids for the EIT problem with full boundary measurements are refined near the boundary and staggered, we can conclude the same abou... |

4 |
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Citation Context ... the distortion of the reconstruction, as shown in the numerical examples below. The conformal mappings Φ and Ψ in decomposition (3.19) are computed numerically using the Schwartz-Christoffel toolbox =-=[19]-=-. In case K = 0 the mapping T is conformal, so we can use (3.5), with the parameters given by (3.9) to obtain T = F. Recall from the numerical results in section 2.5 that the optimal grid depends weak... |

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Citation Context ...conductivities σ1 and σ2, the following estimate holds ∣ ∣ , (1.5) ‖σ1 − σ2‖L∞ (D) ≤ C ∣log ‖Λσ1 − Λσ2‖H1/2 (B)→H−1/2(B) ∣ −α where C and α are positive constants. See also the stability estimates in =-=[32]-=-, that deal with cases where due to noise, the measured data is no longer a DtN map. These estimates are also of logarithmic type. Thus, one needs an exponentially good fit of the data in order to obt... |

3 |
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Citation Context ... ( (F ′ ) T) = I. It means that F is conformal and the resulting conductivity is G∗(σ) = σ ◦ F. (3.4) Since all conformal mappings of the unit disk to itself belong to the family of Möbius transforms =-=[36]-=-, F must be of the form iω z − a F(z) = e , z ∈ D, ω ∈ [0,2π), a ∈ C, |a| < 1, (3.5) 1 − az where we associate R2 with the complex plane C. It remains to determine the constant parameters ω and a in (... |

3 | Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
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(Show Context)
Citation Context ...ortion, due to the artificial anisotropy. This is in fact a consequence of the topology of the grids and resistor networks, and it may be circumvented by considering different topologies, as shown in =-=[39]-=- and in a forthcoming article [10]. We end with the observation that since we proved that the optimal grids for the EIT problem with full boundary measurements are refined near the boundary and stagge... |

3 |
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Citation Context ...isotropic conductivity by a Teichmüller mapping has a uniform anisotropy throughout D. Similar to (3.18), we can define the dilatation of W −1 in terms of a holomorphic function ψ. Then, according to =-=[43]-=-, we can decompose a Teichmüller mapping W into where W = Ψ −1 ◦ AK ◦ Φ, (3.19) ∫ √φ(z)dz, ∫ √ψ(ζ)dζ, Φ(z) = Ψ(ζ) = (3.20) and AK is affine. The mappings Φ and Ψ are conformal away from zeros of φ and... |

2 |
Solving the discrete EIT problem with optimization techniques, 2007. Schlumberger-Doll Report
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Citation Context ...xplicit. The disadvantage is that it quickly becomes unstable, as the number of layers grows. The second approach is to solve the discrete EIT problem with nonlinear, regularized least squares, as in =-=[9]-=-. In general, it is unclear how to regularize the least squares for network recovery using penalty terms, because we cannot speak of regularity assumptions (such as total variation) in the discrete se... |

2 |
Extremal plane quasiconformal mappings with given boundary values
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Citation Context ...nter of the accessible boundary. Our construction of the extremal quasiconformal (Teichmüller) mappings is based on their decomposition in terms of two conformal maps and an affine coordinate stretch =-=[44]-=-, as described in section 3.3. The quasiconformal grids can be used for more general distributions of the measurement points in BA and they have better (more uniform) refinement properties, as shown i... |

1 |
Extremal quasiconformal polygon mappings for arbitrary subdomains of compact Riemann surfaces
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Citation Context ...s of the polygon (in our case eiθk ). The extremal polygonal quasiconformal mapping W takes the boundary points eiθk iτk to prescribed points e , while minimizing the maximum dilatation. According to =-=[46]-=-, the integrals Φ and Ψ in its decomposition map the unit disk conformally onto polygons comprised of a number of rectangular strips. But conformal mappings of the unit disk to polygons are Schwartz-C... |