2,565 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
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
Expectation Propagation for Nonlinear Inverse Problems -- with an Application to Electrical Impedance Tomography
In this paper, we study a fast approximate inference method based on
expectation propagation for exploring the posterior probability distribution
arising from the Bayesian formulation of nonlinear inverse problems. It is
capable of efficiently delivering reliable estimates of the posterior mean and
covariance, thereby providing an inverse solution together with quantified
uncertainties. Some theoretical properties of the iterative algorithm are
discussed, and the efficient implementation for an important class of problems
of projection type is described. The method is illustrated with one typical
nonlinear inverse problem, electrical impedance tomography with complete
electrode model, under sparsity constraints. Numerical results for real
experimental data are presented, and compared with that by Markov chain Monte
Carlo. The results indicate that the method is accurate and computationally
very efficient.Comment: Journal of Computational Physics, to appea
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
High-order regularized regression in Electrical Impedance Tomography
We present a novel approach for the inverse problem in electrical impedance
tomography based on regularized quadratic regression. Our contribution
introduces a new formulation for the forward model in the form of a nonlinear
integral transform, that maps changes in the electrical properties of a domain
to their respective variations in boundary data. Using perturbation theory the
transform is approximated to yield a high-order misfit unction which is then
used to derive a regularized inverse problem. In particular, we consider the
nonlinear problem to second-order accuracy, hence our approximation method
improves upon the local linearization of the forward mapping. The inverse
problem is approached using Newton's iterative algorithm and results from
simulated experiments are presented. With a moderate increase in computational
complexity, the method yields superior results compared to those of regularized
linear regression and can be implemented to address the nonlinear inverse
problem
Tomographic imaging and scanning thermal microscopy: thermal impedance tomography
The application of tomographic imaging techniques developed for medical applications to the data provided by the scanning thermal microscope will give access to true three-dimensional information on the thermal properties of materials on a mm length scale. In principle, the technique involves calculating and inverting a sensitivity matrix for a uniform isotropic material, collecting ordered data at several modulation frequencies, and multiplying the inverse of the matrix with the data vector. In practice, inversion of the matrix in impractical, and a novel iterative technique is used. Examples from both simulated and real data are given
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