Electrical capacitance tomography (ECT) attempts to image the permittivity distribution of an object by measuring the electrical capacitances between sets of electrodes placed around its periphery. Image reconstruction in ECT is a nonlinear and ill-posed inverse problem. Although reconstruction techniques based on a linear approximation are fast, they are not adequate for all cases. In this paper, we study the nonlinearity of the inverse permittivity problem of ECT. A regularized Gauss-Newton scheme has been implemented for nonlinear image reconstruction. The forward problem has been solved at each iteration using the finite element method and the Jacobian matrix is recalculated using an efficient adjoint field method. Regularization techniques are required to overcome the ill-posedness: smooth generalized Tikhonov regularization for the smoothly varying case, and total variation (TV) regularization when there is a sharp transition of the permittivity have been used. The reconstruction results for experimental ECT data demonstrate the advantage of TV regularization for jump changes, and show improvement of the image quality by using nonlinear reconstruction methods.
Soleimani, M., & Lionheart, W. R. B. (2005). Nonlinear image reconstruction for electrical capacitance tomography experimental data using. Measurement Science and Technology, 16(10), 1987-1996. https://doi.org/10.1088/0957-0233/16/10/014