921 research outputs found
EIT Reconstruction Algorithms: Pitfalls, Challenges and Recent Developments
We review developments, issues and challenges in Electrical Impedance
Tomography (EIT), for the 4th Workshop on Biomedical Applications of EIT,
Manchester 2003. We focus on the necessity for three dimensional data
collection and reconstruction, efficient solution of the forward problem and
present and future reconstruction algorithms. We also suggest common pitfalls
or ``inverse crimes'' to avoid.Comment: A review paper for the 4th Workshop on Biomedical Applications of
EIT, Manchester, UK, 200
An Analysis of Finite Element Approximation in Electrical Impedance Tomography
We present a finite element analysis of electrical impedance tomography for
reconstructing the conductivity distribution from electrode voltage
measurements by means of Tikhonov regularization. Two popular choices of the
penalty term, i.e., -norm smoothness penalty and total variation
seminorm penalty, are considered. A piecewise linear finite element method is
employed for discretizing the forward model, i.e., the complete electrode
model, the conductivity, and the penalty functional. The convergence of the
finite element approximations for the Tikhonov model on both polyhedral and
smooth curved domains is established. This provides rigorous justifications for
the ad hoc discretization procedures in the literature.Comment: 20 page
Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
In this work we propose and analyze a numerical method for electrical
impedance tomography of recovering a piecewise constant conductivity from
boundary voltage measurements. It is based on standard Tikhonov regularization
with a Modica-Mortola penalty functional and adaptive mesh refinement using
suitable a posteriori error estimators of residual type that involve the state,
adjoint and variational inequality in the necessary optimality condition and a
separate marking strategy. We prove the convergence of the adaptive algorithm
in the following sense: the sequence of discrete solutions contains a
subsequence convergent to a solution of the continuous necessary optimality
system. Several numerical examples are presented to illustrate the convergence
behavior of the algorithm.Comment: 26 pages, 12 figure
On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification
We present a new approach to convexification of the Tikhonov regularization
using a continuation method strategy. We embed the original minimization
problem into a one-parameter family of minimization problems. Both the penalty
term and the minimizer of the Tikhonov functional become dependent on a
continuation parameter.
In this way we can independently treat two main roles of the regularization
term, which are stabilization of the ill-posed problem and introduction of the
a priori knowledge. For zero continuation parameter we solve a relaxed
regularization problem, which stabilizes the ill-posed problem in a weaker
sense. The problem is recast to the original minimization by the continuation
method and so the a priori knowledge is enforced.
We apply this approach in the context of topology-to-shape geometry
identification, where it allows to avoid the convergence of gradient-based
methods to a local minima. We present illustrative results for magnetic
induction tomography which is an example of PDE constrained inverse problem
The Factorization method for three dimensional Electrical Impedance Tomography
The use of the Factorization method for Electrical Impedance Tomography has
been proved to be very promising for applications in the case where one wants
to find inhomogeneous inclusions in a known background. In many situations, the
inspected domain is three dimensional and is made of various materials. In this
case, the main challenge in applying the Factorization method consists in
computing the Neumann Green's function of the background medium. We explain how
we solve this difficulty and demonstrate the capability of the Factorization
method to locate inclusions in realistic inhomogeneous three dimensional
background media from simulated data obtained by solving the so-called complete
electrode model. We also perform a numerical study of the stability of the
Factorization method with respect to various modelling errors.Comment: 16 page
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
A Partially Reflecting Random Walk on Spheres Algorithm for Electrical Impedance Tomography
In this work, we develop a probabilistic estimator for the voltage-to-current
map arising in electrical impedance tomography. This novel so-called partially
reflecting random walk on spheres estimator enables Monte Carlo methods to
compute the voltage-to-current map in an embarrassingly parallel manner, which
is an important issue with regard to the corresponding inverse problem. Our
method uses the well-known random walk on spheres algorithm inside subdomains
where the diffusion coefficient is constant and employs replacement techniques
motivated by finite difference discretization to deal with both mixed boundary
conditions and interface transmission conditions. We analyze the global bias
and the variance of the new estimator both theoretically and experimentally. In
a second step, the variance is considerably reduced via a novel control variate
conditional sampling technique
Hybrid topological derivative-gradient based methods for nondestructive testing
This paper is devoted to the reconstruction of objects buried in a medium and their material properties by hybrid topological derivative-gradient based methods. After illustrating the techniques in time-harmonic acoustic problems with different boundary conditions and in electrical impedance tomography problems with continuous Neumann conditions, we extend the hybrid method for a realistic model in tomography where the boundary conditions are given at a discrete set of electrodes
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