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Regularized D-bar method for the inverse conductivity problem

Profile image of Jennifer MuellerJennifer Mueller

2009, Inverse Problems & Imaging

https://doi.org/10.3934/IPI.2009.3.599
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Abstract

A strategy for regularizing the inversion procedure for the twodimensional D-bar reconstruction algorithm based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral equation and the scattering transform. It is shown that this leads to a bound on the error in the scattering transform and a stable reconstruction of the conductivity; an explicit rate of convergence in appropriate Banach spaces is derived as well. Numerical results are also included, demonstrating the convergence of the reconstructed conductivity to the true conductivity as the noise level tends to zero. The results provide a link between two traditions of inverse problems research: theory of regularization and inversion methods based on complex geometrical optics. Also, the procedure is a novel regularized imaging method for electrical impedance tomography.

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    Ariungerel Jargal

    Computational and Mathematical Methods in Medicine

    This paper proposes a deep learning method based on electrical impedance tomography (EIT) to estimate the thickness of abdominal subcutaneous fat. EIT for evaluating the thickness of abdominal subcutaneous fat is an absolute imaging problem that aims at reconstructing conductivity distributions from current-to-voltage data. Existing reconstruction methods based on EIT have difficulty handling the inherent drawbacks of strong nonlinearity and severe ill-posedness of EIT; hence, absolute imaging may not be possible using linearized methods. To handle nonlinearity and ill-posedness, we propose a deep learning method that finds useful solutions within a restricted admissible set by accounting for prior information regarding abdominal anatomy. We determined that a specially designed training dataset used during the deep learning process significantly reduces ill-posedness in the absolute EIT problem. In the preprocessing stage, we normalize current-voltage data to alleviate the effects o...

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    Iterative Sinc−convolution method for solving planar D-bar equation with application to EIT
    Ahmad R. Naghsh-Nilchi

    International Journal for Numerical Methods in Biomedical Engineering, 2012

    A new and efficient sinc-convolution algorithm is introduced for the numerical solution of the radiosity equation. This equation has many applications including the production of photorealistic images. The method of sinc-convolution is based on using collocation to replace multi-dimensional convolution-type integrals-such as two dimensional radiosity integral equations-by a system of algebraic equations. The developed algorithm solves for the illumination of a surface or a set of surfaces when both reflectivity and emissivity of those surfaces are known. It separates the radiosity equation's variables to approximate its solution. The separation of variables allows the elimination of the formulation of huge full matrices and therefore reduces required storage, as well as computational complexity, as compared with classical approaches. Also, the highly singular nature of the kernel, which results in great difficulties using classical numerical methods, poses absolutely no difficulties using sinc-convolution. In addition, the new algorithm can be readily adapted for parallel computation for an even faster computational speed. The results show that the developed algorithm clearly reveals the color bleeding phenomenon which is a natural phenomenon not revealed by many other methods. These advantages should make real-time photorealistic image production feasible.

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    Sparse Reconstruction of Log-Conductivity in Current Density Impedance Tomography
    rohit mishra

    Journal of Mathematical Imaging and Vision, 2019

    A new non-linear optimization approach is proposed for the sparse reconstruction of log-conductivities in current density impedance imaging. This framework comprises of minimizing an objective functional involving a least squares fit of the interior electric field data corresponding to two boundary voltage measurements, where the conductivity and the electric potential are related through an elliptic PDE arising in electrical impedance tomography. Further, the objective functional consists of a L 1 regularization term that promotes sparsity patterns in the conductivity and a Perona-Malik anisotropic diffusion term that enhances the edges to facilitate high contrast and resolution. This framework is motivated by a similar recent approach to solve an inverse problem in acousto-electric tomography. Several numerical experiments and comparison with an existing method demonstrate the effectiveness of the proposed method for superior image reconstructions of a wide-variety of log-conductivity patterns.

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    Characterization for stability in planar conductivities
    Daniel Faraco

    Journal of Differential Equations, 2018

    We find a complete characterization for sets of uniformly strongly elliptic and isotropic conductivities with stable recovery in the L 2 norm when the data of the Calderón Inverse Conductivity Problem is obtained in the boundary of a disk and the conductivities are constant in a neighborhood of its boundary. To obtain this result, we present minimal a priori assumptions which turn out to be sufficient for sets of conductivities to have stable recovery in a bounded and rough domain. The condition is presented in terms of the integral moduli of continuity of the coefficients involved and their ellipticity bound as conjectured by Alessandrini in his 2007 paper, giving explicit quantitative control for every pair of conductivities.

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