Limited Electrodes Models in Electrical Impedance Tomography Reconstruction

Abstract

We state the Limited Electrode problem in Electrical Impedance Tomography, and propose solutions inspired by the application of compressed sensing techniques and deep learning strategies on the raw boundary impedance data. These strategies allow to recover the target reconstruction quality while using a relatively low number of nonlinear measurements, assuming sparsity-gradient conductivity. This would help reducing modelling costs and computational power, thus enhancing applicability of EIT.

7.1 Appendix: regularized Gauss-Newton method for the inverse EIT problem

Given a nonlinear residual $r(\sigma )$, the ill-posedness of problem (1) can be alleviated by adding to the cost function a regularization term $R(\sigma )$, which stabilizes the solution

$$\begin{aligned} \sigma ^* \, \in \, \arg \min _{\sigma \in {\mathbb R}^{n_T}} \Vert r(\sigma ) \Vert _2^2+ \frac{\alpha }{2} R(\sigma ), \end{aligned}$$

(7)

with $\alpha >0$ the regularization parameter. An approximated solution of the unconstrained regularized nonlinear minimization problem (7) can be obtained by applying the iterative Regularized Gauss-Newton (RGN) method, which, starting from an initial guess $\sigma ^{(0)}$, performs a line search along the direction $p^{(k)}$ to obtain the new conductivity iterate $\sigma ^{(k+1)}= \sigma ^{(k)} + p^{(k)}$, following the search direction $p^{(k)}$ from the current iterate.

For the choice $R(\sigma ):= \Vert \nabla \sigma \Vert _2^2$, the functional in (7) is the classical generalized nonlinear Tikhonov model, a standard choice in EIT [4], and the (RGN) method can be applied with $p^{(k)}$ determined by solving the linear normal equations

$$\begin{aligned} ( J(\sigma ^{(k)})^T J(\sigma ^{(k)}) + \alpha L^TL)p&= J(\sigma ^{(k)})^T r(\sigma ) - \alpha L^TL \sigma ^{(k)}, \end{aligned}$$

(RGN-Tik)

where $L^TL \in {\mathbb R}^{n_T \times n_T}$ denotes the Laplacian. In case the total variation regularizer $R(\sigma )=\Vert \nabla \sigma \Vert _1$ is considered, we can resort to the discretization presented in [4], with $R(\sigma ) \approx D^TE^{-1}D$, with D denoting the discrete gradient, and $E=1/diag(\sqrt{(D\sigma )^2+\epsilon ^2})$, $\bar{D}=\sqrt{E} D$. The direction $p^{(k)}$ is then obtained by solving the linear normal equations

$$\begin{aligned} ( J(\sigma ^{(k)})^T J(\sigma ^{(k)}) + \alpha \bar{D}^T\bar{D} )p&= J(\sigma ^{(k)})^T r(\sigma ) - \alpha \bar{D}^T\bar{D} \sigma ^{(k)}. \end{aligned}$$

(RGN-TV)

The linear systems in RGN-Tik and RGN-TV are solved by a few iterations of the conjugate gradient method.