Abstract This work presents a new constructive uniqueness proof for Calderón’s inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call piecewise constant layered conductivities (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that “holes in inclusions” or “inclusions-within-inclusions” cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.

中文翻译：

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电阻抗断层扫描中分段恒定分层电导率的重建

摘要 这项工作为 Calderón 的电阻抗断层扫描逆问题提供了一种新的构造唯一性证明，受局部柯西数据的约束，对于一大类我们称为分段常数分层电导率 (PCLC) 的分段常数电导率。由此产生的重建方法仅依赖于局部 Neumann-to-Dirichlet 映射的物理直观单调性原理，因此该方法非常适合有效的数值实现和电极模型的泛化。对于夹杂物检测的相关问题，存在几种直接重建方法，但它们都具有无法确定“夹杂物中的孔”或“夹杂物内夹杂物”的特性。其中一种方法是 Harrach、Seo 和 Ullrich 的单调性方法，事实上，这里介绍的方法是克服这个问题的单调性方法的改进变体。更准确地说，所提出的方法滥用了 PCLC 类型的电导率可以分解为正和/或负扰动的嵌套层，这些层可以通过单调性方法逐层确定。每层的电导率值是通过受单调关系约束的基本一维优化问题找到的。