Determination of a Small Elliptical Anomaly in Electrical Impedance Tomography using Minimal Measurements

Abstract

We consider the problem of determining a small elliptical conductivity anomaly in a unit disc from boundary measurements. The conductivity of the anomaly is assumed to be a small perturbation from the constant background. A measurement of voltage across two point-electrodes on the boundary through which a constant current is passed. We further assume the limiting case when the distance between two electrodes go to zero, creating a dipole field. We show that three such measurements suffice to locate the anomaly size and location inside the disc. Two further measurements are needed to obtain the aspect ratio and the orientation of the ellipse. The investigation includes the studies of the stability of the inverse problem and optimal experiment design.

Figures (7)

• Figure 1: In this schematic, for the case of a source-sink pair, $A$ is the current source and $B$ is the current sink. $A$ is located at coordinates $(\cos\phi,\sin\phi)$ and $B$ is at $(\cos(\phi+\Delta \phi),\sin(\phi+\Delta\phi))$. The conductivity anomaly is represented by the ellipse $D$. The dipole source is the limiting source-sink pair when $\Delta\phi\rightarrow 0$ (modulo scaling). We say that in this case, the dipole is located at $A$. The inverse problem is to determine the geometric properties of the anomaly from measurements of the tangential derivative of the perturbational voltage at $A$. To obtain sufficient information, measurements will be done for several angles $\phi_j$.
• Figure 2: The dipole potential $U_0$ (Left) and the log-plot of the corresponding kernel $|\nabla U_0|^2$ (Right) for a dipole located at $(1,\pi/6)$.
• Figure 3: The measurement points $(\cos\phi_1,\sin\phi_1)$, $(\cos\phi_2,\sin\phi_2)$ and $(\cos\phi_3,\sin\phi_3)$ are indicated by $1$, $2$, and $3$. The circle corresponding to the data $r_1$ is centered at $C_1$ with radius is $\rho_1$, and that corresponding to data $r_2$ is centered at $C_2$ with radius $\rho_2$. These are indicated in blue. The intersection of these two circles locate the ellipse center $(b_1,b_2)$.
• Figure 4: The surface shows the utility function of symmetric designs, where $\phi:=\phi_3$ and $\psi:=\phi_3-\phi_1 = \phi_2-\phi_3$. The maximum value of the utility function $U(\Phi)\approx45.0$ is reached when $\phi\approx36.3^\circ,\psi\approx44.1^\circ$. And the minimum value 2.4 is reached when $\phi\approx218.1^\circ, \psi\approx0.7^\circ$.
• Figure 5: Posteriors of the parameters for the optimal and the worst experiments obtained by Metropolis-Hastings algorithm. The left column is from the best design when $\phi=36.3^\circ, \psi=44.1^\circ$ and the right column is from the worst design when $\phi=218.1^\circ, \psi=0.7^\circ$. The three rows represent the posterior distribution of the area $A$, the $x$ coordinate $b_1$ and the $y$ coordinate $b_2$. The tightness of the distribution can be clearly seen.

Theorems & Definitions (6)

• Remark
• Remark
• Theorem 1
• proof
• Theorem 2
• Remark