D-bar Method

The D-bar method is a direct, non-iterative reconstruction algorithm for 2D EIT. It is based on Nachman’s (1996) constructive uniqueness proof (see Reconstruction Algorithms with Guarantees).

Steps (identifying ${\mathbb{R}}^2\cong \mathbb{C}$, spectral parameter $k\in \mathbb{C}$):

1. From the measured DtN map, compute the traces of the Complex Geometrical Optics Solutions $\psi (\cdot ,k)$ on $\partial \Omega$ by solving a boundary integral equation.
2. Compute the non-physical scattering transform $\mathbf{t}(k)={\int }_{\partial \Omega }{e}^{i\bar{k}\bar{z}}({\Lambda }_{\gamma }-{\Lambda }_1)\psi (\cdot ,k)\mathrm{d}s.$
3. For each $z\in \Omega$, solve the $\bar{\partial }$-equation in $k$ ${\bar{\partial }}_k\mu (z,k)=\frac{\mathbf{t}(k)}{4\pi \bar{k}}{e}_{-k}(z)\overline{\mu (z,k)},e_k(z)=e^{i(kz+\bar{k}\bar{z})},$ with $\mu (z,\cdot )\to 1$ as $|k|\to \infty$.
4. Recover $\gamma (z)=\mu (z,0{)}^2$.

Regularisation. With noisy data, $\mathbf{t}(k)$ is only reliable for small $|k|$. Truncating to $|k|\le R$, with $R$ chosen according to the noise level, gives a provably convergent regularisation strategy (Knudsen et al. 2009). The result is a smoothed (low-pass) conductivity.

Variants. Approximations such as ${\mathbf{t}}^{\exp }$ replace step 1 with a Born-type approximation. Deep D-bar (Hamilton & Hauptmann 2018) post-processes the blurry D-bar image with a U-Net to sharpen edges (see Deep Learning for EIT).

References

1. A. I. Nachman (1996). Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem. Ann. of Math. 143(1), 71–96. doi:10.2307/2118653
2. S. Siltanen, J. Mueller, D. Isaacson (2000). An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem. Inverse Problems 16(3), 681–699. doi:10.1088/0266-5611/16/3/310
3. K. Knudsen, M. Lassas, J. L. Mueller, S. Siltanen (2009). Regularized D-bar method for the inverse conductivity problem. Inverse Probl. Imaging 3(4), 599–624. doi:10.3934/ipi.2009.3.599
4. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344
5. S. J. Hamilton, A. Hauptmann (2018). Deep D-Bar: Real-Time Electrical Impedance Tomography Imaging With Deep Neural Networks. IEEE Trans. Med. Imaging 37(10), 2367–2377. doi:10.1109/TMI.2018.2828303