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

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The work of NH and JT was supported by the Academy of Finland (decision 267789). The work of LP was supported by Estonian government grant PUT1093

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  • We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies to the open upper half-plane. Second, in the unit disk we may choose a region of interest that is magnified using a suitable Möbius transform. To facilitate the efficient use of conformal maps, we introduce input current patterns that are named conformally transformed truncated Fourier basis; in practice, their use corresponds to positioning the available electrodes close to the region of interest. These ideas are numerically tested using simulated continuum data in bounded domains and simulated point electrode data in the half-plane. The connections to practical electrode measurements are also discussed.

    Mathematics Subject Classification: Primary: 65N21, 35R30; Secondary: 65N15.


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  • 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)

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