Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps

Figure 1.  Top left: The real part of a FEM approximation for the solution $u$ to (3) in a homogeneous rectangular domain $\Omega $ with the boundary current density $\varphi_8$ of (33) corresponding to a Schwarz-Christoffel map $\Phi: \Omega \to D$. Top right: The real part of $\tilde{u} = u \circ \Psi$ in $D$. Bottom left: The real part of the true current density $\varphi_8$ as a function of the arclength parameter on $\partial \Omega $. Bottom right: The real part of the virtual current density $\tilde{\phi}_8$ as a function of the polar angle on $\partial D$

Figure 2.  Left: The true conductivity, with the blue area representing the ROI that contains two black anomalies. The red dot indicates the Möbius parameter $a = 0.6$. Right: The conformally mapped conductivity $\tilde{\sigma} = \sigma \circ \mathcal{M}_a^{-1}$, with the image of the ROI under $\mathcal{M}_a$ presented by blue color

Figure 3.  Top row: the target conductivity in three different domains. Bottom row: the virtual conducitivities, i.e., the target conductivities transformed to the unit disk by a Schwarz-Christoffel map that fixes the origin.

Figure 4.  Numerical approximations for the real part of the truncated scattering transform $\mathbf{t}_{R, c}$ of (24) for the five configurations in Figure 3 with $c = 10$. The white parts correspond to $|\mathbf{t}|>c$ or $|k|>R$, i.e, $\mathbf{t}_{R, c} = 0$

Figure 5.  Reconstructions of the test conductivity in three different domains and the corresponding relative $L^2(\Omega )$-errors. Top row: the target configurations. Middle row: reconstructions by the D-bar method using the current patterns (19) with $N = 16$. Bottom row: reconstructions by the D-bar method in the virtual domain (i.e., the unit disk) using the conformally transformed current patterns (33) with $N = 16$ in the true domain

Figure 6.  Magnification of a ROI. Top row: the true target configurations. Middle row: the conformally transformed conductivities in the virtual domain with magnified ROIs. On their boundaries the point corresponding to $s = 0$ is marked with a green dot. The black dot close to the ROI is mapped to the origin. Bottom row: The real parts of the conformally transformed Fourier current pattern $\varphi_{16}$ of (33) for the respective true target domains

Figure 7.  Reconstructions by magnifying the ROIs (cf. Figure 6) with the corresponding relative $L^2$-errors over the ROIs. Top row: the ROIs in the original domains of Figure 6. Middle row: reconstructions by the D-bar method using the current patterns (19) with $N = 16$. Bottom row: reconstructions by the D-bar method in the virtual domain (i.e., the unit disk) using the conformally transformed current patterns (33) with $N = 16$ in the true domain. The left column corresponds to noiseless data, whereas the reconstructions in the right column are based on measurements corrupted by $1\%$ of additive noise

Figure 8.  Conductivity inhomogeneities in the upper half-plane (left) and the corresponding virtual conductivities in the unit disk corresponding to $\mathcal{M}_b$ with five different values for $b$ (right). The black dots indicate the sampling points for the boundary potentials corresponding to the (truncated) currents (33), with $N = 5$, on the measurement interval $[-2.5+b, 2.5+b]$. These dots are mapped to an (incomplete) equidistant grid of points on the unit circle by $\mathcal{M}_b$. The gray areas on the right correspond to the exterior of the rectangular subset considered on the left

Figure 9.  D-bar reconstruction in the upper half-plane obtained by using five different Möbius transforms, i.e., $\mathcal{M}_b$ with $b = -3, -1.5, 0, 1.5, 3$, for mapping the half-plane onto the unit disk. The vertical line segments indicate the used values of $b$ (cf. Figure 8)