  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.
