The Kernel Method for Electrical Resistance Tomography (2312.05059v2)

Abstract: This paper treats the inverse problem of retrieving the electrical conductivity starting from boundary measurements in the framework of Electrical Resistance Tomography (ERT). In particular, the focus is on non-iterative reconstruction methods that are compatible with real-time applications. In this work, the Kernel Method, a new non-iterative reconstruction method for Electrical Resistance Tomography, is presented. The imaging algorithm deals with the problem of retrieving one or more anomalies of arbitrary shape, topology, and size, embedded in a known background (inverse obstacles problem). The foundation of the Kernel Method is that if there exists a proper current density applied at the boundary (Neumann data) of the domain that is able to produce the same measurements with and without the anomaly, then this boundary source produces a power density that vanishes in the region occupied by the anomaly, when applied to the problem involving the background material only. Therefore, the Kernel Method consists of (i) evaluating a proper current density g at the boundary of the domain of interest, by solving a proper linear eigenvalue problem, (ii) solving one direct problem for the configuration without anomaly and driven by g, and (iii) reconstructing the anomaly as the region in which the power density is negligible. This new tomographic method has a very simple numerical implementation that requires a very low computational cost. In addition to theoretical results, an extensive numerical campaign proves the effectiveness of this new imaging method.