Smoothened Complete Electrode Model

Hyvönen, Nuutti; Mustonen, Lauri

doi:10.1137/17m1124292

Cited by 29 publications

(89 citation statements)

References 31 publications

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“…The mapping C 1 (E, R m ) a → U (a) ∈ R M is Fréchet differentiable at the origin [8,16]. The associated Fréchet derivative in the direction a ∈ C 1 (E, R m ) can be assembled using…”

Section: 3mentioning

confidence: 99%

“…We choose the corresponding prior covariance to be Γ ζ = (2ς ζζ /τ ζ ) 2 I, wherẽ τ ζ andς are the expected value and standard deviation employed when drawing the target contact conductance values. The reason for this ad hoc choice is two-fold: First of all, as the contact conductance models for the data simulation and the reconstruction process are different, there is no reason to expect that the distributions of the associated parameters are the same; for further analysis, see [16], where it is noted that the smoothened model often leads to significantly higher peak values for the contact conductance function. Secondly, the estimation of the contact conductances is usually a rather stable process which allows one to employ only mild regularization, i.e., twice as large ratio between the prior standard deviation and expected value than used in the data simulation step.…”

Section: Reconstructionsmentioning

confidence: 99%

“…In the following, an electrode potential pattern U may be treated either as a piecewise constant function supported on E or as an element of C M ; the correct interpretation should be clear from the context. The forward problem corresponding to the smoothened CEM is as follows [16]: the electromagnetic potential u inside Ω and the piecewise constant electrode potential U weakly satisfy ∇ · (σ∇u) = 0…”

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confidence: 99%

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Computational Framework for Applying Electrical Impedance Tomography to Head Imaging

Candiani¹,

Hannukainen²,

Hyvönen³

2019

SIAM J. Sci. Comput.

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This work introduces a computational framework for applying absolute electrical impedance tomography to head imaging without accurate information on the head shape or the electrode positions. A library of fifty heads is employed to build a principal component model for the typical variations in the shape of the human head, which leads to a relatively accurate parametrization for head shapes with only a few free parameters. The estimation of these shape parameters and the electrode positions is incorporated in a regularized Newton-type output least squares reconstruction algorithm. The presented numerical experiments demonstrate that strong enough variations in the internal conductivity of a human head can be detected by absolute electrical impedance tomography even if the geometric information on the measurement configuration is incomplete to an extent that is to be expected in practice.

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Section: 3mentioning

confidence: 99%

Section: Reconstructionsmentioning

confidence: 99%

mentioning

confidence: 99%

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Computational Framework for Applying Electrical Impedance Tomography to Head Imaging

Candiani¹,

Hannukainen²,

Hyvönen³

2019

SIAM J. Sci. Comput.

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“…standard shift theorem yields the H 2 (Ω) regularity of u, under some regularity assumptions on Γ Diss and Γ Dir since ξu belongs to H 1/2 (Γ Diss ). With such an approach we obtain the estimate u 2,Ω f + ξu 1 2 ,ΓDiss , and using (11) we conclude that…”

Section: Remark 22mentioning

confidence: 90%

“…for some positive constant d . Such a problem occur in different applications like complete electrode models [7,10,11,16], singularly perturbed radiation problem [6,18], embedding of quantum structures mechanically described into a macroscopic flow [12].…”

mentioning

confidence: 99%

Uniform a priori estimates for elliptic problems with impedance boundary conditions

Chaumont-Frelet¹,

Nicaise²,

Tomezyk³

2020

Communications on Pure &Amp; Applied Analysis

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We derive stability estimates in H 2 for elliptic problems with impedance boundary conditions that are uniform with respect to the impedance coefficient. Such estimates are of importance to establish sharp error estimates for finite element discretizations of contact impedance and high-frequency Helmholtz problems. Though stability in H 2 is easily obtained by employing a "bootstrap" argument and well-established result for the corresponding Neumann problem, this strategy leads to a stability constant that increases with the impedance coefficient. Here, we propose alternative proofs to derive sharp and uniform stability constants for domains that are convex or smooth.

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Generalized linearization techniques in electrical impedance tomography

Hyvönen

Mustonen

2018

Numer. Math.

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Electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current-voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such. In this paper, we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity. It is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations. Inspired by these observations, we present and analyze a new linearization technique which is based on the logarithm of the Neumannto-Dirichlet operator. The method is directly applicable to discrete settings, including the complete electrode model. We also consider Fréchet derivatives of the logarithmic operators. Numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography.n. hyvönen and l. mustonen 2 respect to the conductivity can be computed explicitly [4,14,18]. By applying the chain rule of Banach spaces, or via direct computation, it is possible to consider derivatives with respect to the resistivity or the logarithm of the conductivity as well. In this paper, we compare the linearization errors resulting from these different input parametrizations of the electrical properties. We also discuss the corresponding methods for the complete electrode model (CEM), which is an accurate model for practical EIT measurements [5,21].As the main novelty, we introduce the logarithm of the Neumann-to-Dirichlet operator and use its differentiability properties to construct a new linearization method for EIT. Loosely speaking, the traditional forward operator is replaced by an operator that maps the logarithm of the conductivity to the logarithm of the boundary potentials, the latter being understood in a sense of linear operators. The corresponding least squares inversion algorithm then fits the computed logarithmic potentials to the logarithm of the measurement matrix. Although this logarithmic forward operator is still nonlinear, numerical experiments show that the resulting linearization errors are in most cases smaller than with any other considered linearization method.This paper is organized as follows. In Section 2, the continuum forward model of EIT and the related Neumann-to-Dirichlet operator are reviewed. We write the dependence of the Neumannto-Dirichlet operator on the electrical properties in three different ways and recall also the corresponding Fréchet derivatives. Section 3 presents the formal definition for the logarithm of the Neumann-to-Dirichlet operator and studies its differentiability and other properties as an unbounded operator on the space of square-integ...

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Smoothened Complete Electrode Model

Cited by 29 publications

References 31 publications

Computational Framework for Applying Electrical Impedance Tomography to Head Imaging

Computational Framework for Applying Electrical Impedance Tomography to Head Imaging

Uniform a priori estimates for elliptic problems with impedance boundary conditions

Generalized linearization techniques in electrical impedance tomography

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