Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations
Comparison of linear and non-linear monotonicity-based shape reconstruction using exact matrix characterizations
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DOI：
10.1080/17415977.2017.1290088
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发表时间：
2016-02
影响因子：
1.3
通讯作者：
Henrik Garde
中科院分区：
文献类型：
--
作者：
Henrik Garde
检测电导率中的不均匀性是电阻抗断层成像中的逆问题的特殊情况，其导致快速直接重建方法。一种这样的方法可以在合理的假设下，基于Neumann-to-Dirichlet映射（非线性）或其Fréchet导数（线性）的单调性特性精确地表征不均匀性。我们给出了一个比较的非线性和线性的方法在测量噪声的存在下，并数值表明，这两种方法基本上是相同的重建在单位磁盘域。为了公平的比较，在探测单调性关系时使用精确的矩阵特征，以避免从数值解到偏微分方程和数值积分的误差。使用Neumann-to-Dirichlet映射的特殊分解也使得非线性方法在单位圆盘几何中与线性方法一样快。
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 Fréchet 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.