A Bayesian filtering approach to layer stripping for electrical impedance tomography
A Bayesian filtering approach to layer stripping for electrical impedance tomography
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DOI：
10.1088/1361-6420/ab6f9e
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发表时间：
2020-01
期刊：
影响因子：
2.1
通讯作者：
D. Calvetti;S. Nakkireddy;E. Somersalo
中科院分区：
文献类型：
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作者：
D. Calvetti;S. Nakkireddy;E. Somersalo
层剥离法是一种求解椭圆型偏微分方程逆边值问题的方法，最初在文献中提出用于求解电阻抗断层成像（EIT）的Calderón问题，其中数据由边界上的Neumann-to-Dirichlet算子组成。在边界附近定义一个切线法向坐标系，将数据推广到物体内部切向表面上的一族边界算子，并证明了这些算子关于法向坐标满足一个非线性Riccati型微分方程。层剥离过程由两个交替步骤的序列组成：从边界数据的空间高频极限估计当前边界附近的电导率，并且通过Riccati方程将边界算子通过薄层进一步传播到域中。这样，从边界开始并向内逐层估计域内部的未知电导率。EIT问题的不适定性表现在向后Riccati方程对边界数据中的误差的高度敏感性，从而导致解在有限时间内爆炸，因此需要正则化。在这篇文章中，我们制定了层剥离过程中的贝叶斯逆问题的框架，我们重新审视的实施贝叶斯过滤。更具体地说，我们重铸相关的逆边值问题作为一个状态估计问题，并提出了一种算法，其数值解的基础上集合卡尔曼滤波（EnKF）。我们提出的新贝叶斯层剥离方法是相当强大的，衍生的自由和本质上适合量化的不确定性估计。此外，我们表明，该算法可以扩展到现实的数据收集，通过使用有限数量的接触电极。
Layer stripping is a method for solving inverse boundary value problems for elliptic PDEs, originally proposed in the literature for solving the Calderón problem of electrical impedance tomography (EIT), where the data consist of the Neumann-to-Dirichlet operator on the boundary. Defining a tangent–normal coordinate system near the boundary, the data are extended to a family of boundary operators on tangential surfaces inside the body, and it is shown that the operators satisfy a non-linear Riccati type differential equation with respect to the normal coordinate. The layer stripping process consists of a sequence of two alternating steps: the conductivity near the current boundary is estimated from the spatial high-frequency limit of the boundary data, and the boundary operator is propagated through a thin layer further into the domain via the Riccati equation. This way, the unknown conductivity in the interior of the domain is estimated layer by layer starting from the boundary and moving inward. The ill-posedness of the EIT problem manifests itself in such high sensitivity of the backwards Riccati equation to errors in the boundary data to cause the solutions to blow up in finite time, thus requiring regularization. In this article, we formulate the layer stripping process in the framework of Bayesian inverse problems, and we revisit the implementation in the light of Bayesian filtering. More specifically, we recast the related inverse boundary value problem as a state estimation problem, and propose an algorithm for its numerical solution based on ensemble Kalman filtering (EnKF). The new Bayesian layer stripping approach that we propose is quite robust, derivative-free and intrinsically suited for the quantification of uncertainties in the estimate. Furthermore, we show that the algorithm can be extended to realistic data collected by using a finite number of contact electrodes.