12 research outputs found
The regularized monotonicity method: detecting irregular indefinite inclusions
In inclusion detection in electrical impedance tomography, the support of
perturbations (inclusion) from a known background conductivity is typically
reconstructed from idealized continuum data modelled by a Neumann-to-Dirichlet
map. Only few reconstruction methods apply when detecting indefinite
inclusions, where the conductivity distribution has both more and less
conductive parts relative to the background conductivity; one such method is
the monotonicity method of Harrach, Seo, and Ullrich. We formulate the method
for irregular indefinite inclusions, meaning that we make no regularity
assumptions on the conductivity perturbations nor on the inclusion boundaries.
We show, provided that the perturbations are bounded away from zero, that the
outer support of the positive and negative parts of the inclusions can be
reconstructed independently. Moreover, we formulate a regularization scheme
that applies to a class of approximative measurement models, including the
Complete Electrode Model, hence making the method robust against modelling
error and noise. In particular, we demonstrate that for a convergent family of
approximative models there exists a sequence of regularization parameters such
that the outer shape of the inclusions is asymptotically exactly characterized.
Finally, a peeling-type reconstruction algorithm is presented and, for the
first time in literature, numerical examples of monotonicity reconstructions
for indefinite inclusions are presented.Comment: 28 pages, 7 figure
Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography
The inverse problem of electrical impedance tomography is severely ill-posed,
meaning that, only limited information about the conductivity can in practice
be recovered from boundary measurements of electric current and voltage.
Recently it was shown that a simple monotonicity property of the related
Neumann-to-Dirichlet map can be used to characterize shapes of inhomogeneities
in a known background conductivity. In this paper we formulate a
monotonicity-based shape reconstruction scheme that applies to approximative
measurement models, and regularizes against noise and modelling error. We
demonstrate that for admissible choices of regularization parameters the
inhomogeneities are detected, and under reasonable assumptions, asymptotically
exactly characterized. Moreover, we rigorously associate this result with the
complete electrode model, and describe how a computationally cheap
monotonicity-based reconstruction algorithm can be implemented. Numerical
reconstructions from both simulated and real-life measurement data are
presented
Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography
The aim of electrical impedance tomography is to reconstruct the admittivity
distribution inside a physical body from boundary measurements of current and
voltage. Due to the severe ill-posedness of the underlying inverse problem, the
functionality of impedance tomography relies heavily on accurate modelling of
the measurement geometry. In particular, almost all reconstruction algorithms
require the precise shape of the imaged body as an input. In this work, the
need for prior geometric information is relaxed by introducing a Newton-type
output least squares algorithm that reconstructs the admittivity distribution
and the object shape simultaneously. The method is built in the framework of
the complete electrode model and it is based on the Fr\'echet derivative of the
corresponding current-to-voltage map with respect to the object boundary shape.
The functionality of the technique is demonstrated via numerical experiments
with simulated measurement data.Comment: 3 figure
Fine-tuning of the Complete Electrode Model
The Complete Electrode Model (CEM) is a realistic measurement model for Electrical Impedance Tomography. We present a non-uniform discretization of the conductivity space based on its sensitivity to boundary data and an adaptive adjustment of electrode parameters leading to improved reconstructions of Newton-type solvers. We demonstrate the performance of this concept when reconstructing with incorrect geometry assumptions from noisy data
Reunanmuotoanalyysin sovelluksia impedanssitomografiaan
A typical electrical impedance tomography (EIT) reconstruction is ruined if the shape of theimaged body or the electrode locations are not accurately known. In this dissertation, a newapproach based on (boundary) shape analysis is presented for online adaptation of themeasurement geometry model.
It is shown that the forward operator of the complete electrode model (CEM) of EIT is Fréchet differentiable with respect to both the outer boundary shape of the object and the electrodelocations. A dual technique allows feasible computation of the gradients in a Newton-type'output least squares' algorithm for the simultaneous reconstruction of the conductivity andthe measurement geometry. Shape calculus techniques are also applied to optimal experimentdesign for EIT. To be more precise, numerical optimization of the electrode positions is carriedout with respect to posterior covariance related criteria derived from the Bayesian inversionparadigm.
Special attention is paid to the Sobolev regularity properties of the CEM essential for the shape analysis. By interpolation of Sobolev spaces it is proven that the CEM is a perturbation of the less regular shunt model of EIT. Consequently, instability in the computation of the numerical shape derivative can be expected if the contact resistances are small.Impedanssitomografiassa (EIT) kappaleen reunanmuodon ja elektrodien sijaintienepätarkka mallinnus pilaa tavallisesti johtavuuden rekonstruktion. Väitöskirjassa esitelläänuusi (reunan)muotoanalyysiin perustuva menetelmä, joka mahdollistaa mittausgeometrianreaaliaikaisen sovittamisen dataan.      Â
Työssä osoitetaan, että EIT:n täydellisen elektrodimallin (CEM) suoran ongelman ratkaisuon FrĂ©chet-derivoituva sekä kappaleen reunanmuodon että elektrodien sijaintien suhteen.Johtavuusjakauma ja mittausgeometria rekonstruoidaan yhtäaikaisesti Newton-tyyppiselläneliösumman minimointialgoritmilla. Tarvittavat gradientit voidaan laskea tehokkaastiduaalitekniikan avulla. Muotoanalyyttisiä menetelmiä sovelletaan myös optimaalisen EIT-mittauksen suunnitteluun, missä elektrodien sijainnit optimoidaan numeerisestiposteriorikovarianssiin liittyvien Bayesiläisten kriteerien mukaisesti.      Â
Erityistä huomiota kiinnitetään CEM:n Sobolev-säännöllisyysominaisuuksiin, jotka ovatolennaisia työssä sovelletun muotoanalyysin kannalta. Sobolev-avaruuksien väliselläinterpolaatiolla osoitetaan, että CEM on perturboitu versio eräästä ideaalisestaelektrodimallista (shunt-malli). Tästä tuloksesta voidaan päätellä muotoderivaattojennumeerisen approksimoinnin epästabiilius, mikäli kontaktiresistanssit ovat pieniä
Construction of invisible conductivity perturbations for the point electrode model in electrical impedance tomography
Peer reviewe
Construction of indistinguishable conductivity perturbations for the point electrode model in electrical impedance tomography
International audienceWe explain how to build invisible isotropic conductivity perturbations of the unit conductivity in the framework of the point electrode model for two-dimensional electrical impedance tomography. The theoretical approach, based on solving a fixed point problem, is constructive and allows the implementation of an algorithm for approximating the invisible perturbations. The functionality of the method is demonstrated via numerical examples
Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography
Peer reviewe