379 research outputs found
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
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
Comparison of linear and non-linear monotononicity-based shape reconstruction using exact matrix characterizations
Detecting inhomogeneities in the electrical conductivity is a special case of
the inverse problem in electrical impedance tomography, that leads to fast
direct reconstruction methods. One such method can, under reasonable
assumptions, exactly characterize the inhomogeneities based on monotonicity
properties of either the Neumann-to-Dirichlet map (non-linear) or its Fr\'echet
derivative (linear). We give a comparison of the non-linear and linear approach
in the presence of measurement noise, and show numerically that the two methods
give essentially the same reconstruction in the unit disk domain. For a fair
comparison, exact matrix characterizations are used when probing the
monotonicity relations to avoid errors from numerical solution to PDEs and
numerical integration. Using a special factorization of the
Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear
method in the unit disk geometry.Comment: 18 pages, 5 figures, 1 tabl
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