In this paper we consider the inverse problem of electrical conductivity retrieval starting from boundary measurements, in the framework of Electrical Resistance Tomography (ERT). In particular, the focus is on non-iterative reconstruction algorithms, compatible with real-time applications. In this work a new non-iterative reconstruction method for Electrical Resistance Tomography, termed Kernel Method, is presented. The imaging algorithm deals with the problem of retrieving the shape of one or more anomalies embedded in a known background. The foundation of the proposed method is given by the idea that if there exists a current flux at the boundary (Neumann data) able to produce the same voltage measurements on two different configurations, with and without the anomaly, respectively, then the corresponding electric current density for the problem involving only the background material vanishes in the region occupied by the anomaly. Coherently with this observation, the Kernel Method consists in (i) evaluating a proper current flux at the boundary $g$, (ii) solving one direct problem on a configuration without anomaly and driven by $g$, (iii) reconstructing the anomaly from the spatial plot of the power density as the region in which the power density vanishes. This new tomographic method has a very simple numerical implementation at a very low computational cost. Beside theoretical results and justifications of our method, we present a large number of numerical examples to show the potential of this new algorithm.

翻译：本文在电阻抗层析成像（ERT）框架下，研究从边界测量值反演电导率分布这一逆问题，重点关注适用于实时应用的非迭代重建算法。本文提出一种名为“核方法”的新型非迭代电阻抗层析成像重建算法。该成像算法旨在恢复嵌入已知背景中的单个或多个异常体的形状。所提方法基于以下核心理念：若存在一种边界电流通量（诺伊曼数据），能分别在有异常和无异常两种不同配置下产生相同的电压测量值，则仅涉及背景材料的无异常问题中对应的电流密度在异常体占据的区域将为零。基于这一发现，核方法包括：（1）评估边界上的合适电流通量$g$；（2）求解一个由$g$驱动的无异常配置下的正问题；（3）通过功率密度的空间分布图，将功率密度为零的区域识别为异常体并重建其形状。该新型层析成像方法数值实现极为简便，计算成本极低。除理论结果与方法论证外，我们通过大量数值算例展示了该新算法的应用潜力。