The Kernel Method for Electrical Resistance Tomography

TL;DR

This paper proposes the Kernel Method (KM), a non-iterative, real-time approach to reconstruct anomalies in Electrical Resistance Tomography from boundary measurements. KM uses the Neumann-to-Dirichlet maps $\Lambda_D$ and $\Lambda_{bg}$ to form the operator $\Delta\Lambda=\Lambda_D-\Lambda_{bg}$ and seeks boundary data $g$ in or near its kernel by analyzing eigenpairs $(\lambda_n,g_n)$ with small $\lambda_n$. For a chosen $g_n$, one solves the background problem to obtain the power density $p_{BG}=\sigma_{bg}|\nabla u_{bg}|^2$ and defines the anomaly as the region where $p_{BG}$ is negligible, with the threshold $\varepsilon^*$ determined from $\lambda_n$ and the measured data; noise is handled via perturbation bounds and a plateau-based selection of the eigenpair. Theoretical results connect small eigenvalues to vanishing current in $D$, and the paper validates KM through analytical examples in concentric geometries and extensive numerical experiments showing robustness to noise, low computational cost, and good reconstruction of multiple or convex anomalies.

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.

Figures (7)

• Figure 1: Perturbation of the eigenvalues of the key operator $\Delta\Lambda$. The domain $\Omega$ is a circle centered in the origin with radius 10, while the anomaly $D$ is a circle centered in the origin and radius 4. The electrical conductivities are $\sigma_{bg}=\qty{200}{\siemens.\m}$ and $\sigma_a=1S.\m$. The norm of the operator $N$ is $\delta$.
• Figure 2: The Kernel Method is applied to retrieve the unknown anomaly $D$.
• Figure 3: Behavior of the reconstructed internal radius $\tilde{r}_i$, with respect to the order of the selected eigenfunction.
• Figure 4: The Kernel Method is applied to retrieve the circular crown $r_1\leq r\leq r_2$.
• Figure 5: Behavior of the reconstructed outer radius $\tilde{r}_2$ of the circular crown, with respect to the order of the selected eigenfunction.

Theorems & Definitions (12)

• Lemma 2.1: see Ge08HaUl13art:ik98
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Theorem 3.3: Weak UCP art:ale12, art:gar86
• Lemma 3.4
• proof
• Remark 3.5
• Proposition 5.1