Numerical solution method for the dbar-equation in the plane

Applied and Computational Harmonic Analysis, 2010

A numerical method is introduced for the evaluation of complex geometrical optics (cgo) solutions to the conductivity equation ∇ • σ ∇u(•, k) = 0 in R 2 for piecewise smooth conductivities σ. Here k is a complex parameter. The algorithm is based on the solution by Astala and Päivärinta (2006) [1] of Calderón's inverse conductivity problem and involves the solution of a Beltrami equation in the plane with an exponential asymptotic condition. The numerical strategy is to solve a related periodic problem using fft and gmres and show that the solutions agree on the unit disc. The cgo solver is applied to the problem of computing nonlinear Fourier transforms corresponding to nonsmooth conductivities. These computations give new insight into the D-bar method for the medical imaging technique of electric impedance tomography. Furthermore, the asymptotic behavior of the cgo solutions as k → ∞ is studied numerically. The evidence so gained raises interesting questions about the best possible decay rates for the subexponential growth argument in the uniqueness proof for Calderón's problem with L ∞ conductivities.

1970

The Electrical Impedance Tomography (BIT) utilizes CPU time-consuming minimization techniques for imaging an object. In this paper we present a computationally fast method for the forward/inverse BIT problem based on an asymptotic solution to Laplace's equation for an imaged domain with a small size inclusion situated inside of it. In this solution the principal term is represented by a dipole term. The computational time required by an BIT problem using the Dipole Approximation (DA) can be two orders of magnitude smaller than when using regular BEM. Results of numerical tests show a substantial improvement in the speed of the inverse problem solution when using the DA BEM approach.

2010

The main objective of this paper is to contribute to the development of a reconstruction method for the electrical impedance tomography (EIT) problem using the boundary element method (BEM). The idea is to adopt BEM to improve the performance of the solution of the direct problem. The EIT problem is generally formulated as an inverse problem and the approach to its solution consists in minimizing an error functional that confronts two different models of the same problem, one calculated from a numerical method and another experimentally obtained. An iterative process is performed until the global minimum (sought image) of the error functional has been reached. Therefore, the direct problem is solved many times to reconstruct the image, and the improvement of its performance is essential to obtain solutions in viable time. BEM can provide this benefit since it effectively reduces the dimension of the numerical problem and, consequently, results in a considerable reduction in the number of variables required for the solution. Tests with the developed BEM algorithm considering a two-dimensional domain with piecewise constant conductivities, synthetic data for the experimental setup and different values and distribution of the contrast have been performed. The numerical results have shown that the method is quite promising.

2000

In this work we present a parallel method for the solution of the inverse problem associated to Electrical Impedance Tomography (EIT). This technique tries to recover the spatial distribution of electrical conductivity in the interior of a body from electric potential measurements taken on the external boundary of the body. The problem we address in this work consists in estimating the shape and position of cavities inside a two-dimensional domain. In the present approach a parametric representation of the geometry of the internal cavities is used and the problem is formulated as an optimization one, where the objective function is the fit between simulated and measured electrical potentials. The simulated electric potentials are numerically obtained by casting the Poison's Equation and applying the Boundary Element Method (BEM) for discretization. As experimental data was not available, the BEM is also used to generate synthetic sets of measured electric potentials. The optimization strategy is based on a hybrid method that uses a Genetic Algorithm (GA) as a localization tool and a classical derivative free mathematical programming routine to enhance the spatial precision achieved by the GA. Since the GA dominates computation time, this part of the method is parallelized in a master-slave fashion. Results showing the performance of the proposed parallel strategy are presented for the standard problem of identifying cylindrical cavities with the influence of noise in the synthetic measurements.

IEEE Transactions on Magnetics, 1998

Although most finite element programs have quite effective iterative solvers such as an incomplete Cholesky (IC) or symmetric successive overrelaxation (SSOR) preconditioned conjugate gradient (CG) method, the solution time may still become unacceptably long for very large systems. Convergence and thus total solution time can be shortened by using better preconditioners such as geometric multigrid methods. Algebraic multigrid methods have the supplementary advantage that no geometric information is needed and can thus be used as black box equation solvers. In case of a finite element solution of a nonlinear magnetostatic problem, the algebraic multigrid method reduces the overall computation time by a factor of 6 compared to a SSOR-CG solver.

IEEE Transactions on Medical Imaging, 1994

Abstmct-A time-harmonic formulation for the electrical impedance tomography (EIT) inverse problem accounting for electrodynamic effects is derived. This work abandons the standard electrostatic impedance model for a full-wave 2'-matrix model The advantage of this method is an accurate physical model that includes linite frequency effects, such as diffusion phenomena, and electrode contact impedance effects. This model offers the potential for increased resolution and larger invertible contrast objects than other methods when used on experimental data, because it may represent a more realistic physical model. Also, an accurate gradient matrix is used in the Newton iterative method so the image reconstruction converges in a few iterations. These advantages are realized with no increase in the computational complexity of this algorithm, compared to the static finite element model. A calibration technique is suggested for measurement systems, to test the validity of a theoretical model that includes electrode contact impedance effects.

Piers Online, 2005

A reconstruction algorithm for three dimensional time harmonic impedance imaging based on the Modified Perturbation Method (MPM), [1], is proposed. Both the object conductivity (σ) and permittivity (ε) are reconstructed. The original Perturbation method developed for static problems was modified in order to apply in time harmonic problem in higher frequency. In this case complex permittivity and complex voltages are involved. So, major modifications have been made in order to achieve accepted results. The jacobian matrix is expressed in complex form and the modified perturbation reconstruction algorithm formulated accordingly. A number of successful reconstructions were carried out for different complex permittivity profiles, but all of them based on a computer phantom approach.

Inverse Problems, 2020

Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed using harmless electric currents. The currents are fed through electrodes placed on the surface of the target, and the data consists of voltages measured at the electrodes resulting from a linearly independent set of current injection patterns. EIT aims to recover the internal distribution of electrical conductivity inside the target. The inverse problem underlying the EIT image formation task is nonlinear and severely ill-posed, and hence sensitive to modeling errors and measurement noise. Therefore, the inversion process needs to be regularized. However, traditional variational regularization methods, based on optimization, often suffer from local minima because of nonlinearity. This is what makes regularized direct (non-iterative) methods attractive for EIT. The most developed direct EIT algorithm is the D-bar method, based on Complex Geometric Optics solutions and a nonlinear Fourier transform. Variants and recent developments of D-bar methods are reviewed, and their practical numerical implementation is explained.

IEEE Transactions on Magnetics, 2000

The transfinite-element time-domain method using the A-formulation for analyzing printed circuit boards is presented in this paper. To solve the resulting system of equations, a multigrid algorithm is implemented. The standard prolongation operator has to be altered to fit the transfinite elements and a new method for constructing the hierarchy of grids is presented. Numerical examples will show the properties of this method.

Electrical impedance tomography (EIT) is a newly developed technique by which impedance measurements from the surface of an object are reconstructed into impedance image. The two-dimensional (2-D) EIT problem is regarded as a simplified model. As 2-D model cannot physically represent the three-dimensional (3-D) structure, the spatial information of the place where the impedance is changed by some diseases cannot be detected accurately. Therefore, 3-D EIT is necessary. In this paper, the finite element method (FEM) of 3-D EIT forward problem is presented. A sphere model is studied. The tetrahedron element is used in the meshing. Two types of sphere model, uniform model and multiplayer model are analyzed. The uniform sphere model, to which the analytical solution is applicable, is used to verify the developed FEM as well as examine the accuracy. The comparison between the numerical solution and the analytical solution shows the correctness of the developed FEM for EIT forward problem. The multiplayer model, four-layer model and three-layer model, is used to investigate the physical potential distribution inside the inhomogeneous sphere model. Reasonable potential distributions are obtained. Index Terms-Electrical impedance tomography (EIT), finite element method (FEM), forward problem, inverse problem.