Formulation of cost functionals for different measurement principles in nonlinear capacitance tomography

A model-based, nonlinear reconstruction for electrical capacitance tomography (ECT) is considered as an inverse problem to find the spatial distributed permittivities in a pipe. The corresponding forward problem is defined by the Laplace equation and depending on the measurement principle two different sets of boundary data can be obtained for the inverse problem. Dirichlet boundary data correspond to voltage measurements and Neumann boundary data to the measurement of charges on the electrodes. For each type of data different cost functionals are defined. In this paper, the influence of the functionals on the inverse problem is examined. Therefore, two different, nonlinear reconstruction methods validate the functionals. One is based on a fixed grid and the forward problem is solved by the finite-element method. Second, a moving contour described by a level set formulation is used to reconstruct the boundaries of different phases. In this case, the field problem is solved by a boundary element formulation. The reconstructions are based on a Gauss–Newton scheme and the gradient is calculated analytically by the material derivative method. Reconstruction results for measurements with an ECT prototype sensor are presented and good results are reported independent of the implemented cost functional.

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