The Kernel Method for Electrical Resistance Tomography

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.