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Nonlinear solution approaches for inverse problems require fast simulation techniques for the underlying sensing problem. In this work, the authors investigate finite element (FE) based sensor simulations for the inverse problem of electrical capacitance tomography. Two known computational bottlenecks are the assembly of the FE equation system as well as the computation of the Jacobian. Here, existing computation techniques like adjoint field approaches require additional simulations. This paper aims to present fast numerical techniques for the sensor simulation and computations with the Jacobian matrix.

For the FE equation system, a solution strategy based on Green’s functions is derived. Its relation to the solution of a standard FE formulation is discussed. A fast stiffness matrix assembly based on an eigenvector decomposition is shown. Based on the properties of the Green’s functions, Jacobian operations are derived, which allow the computation of matrix vector products with the Jacobian for free, i.e. no additional solves are required. This is demonstrated by a Broyden–Fletcher–Goldfarb–Shanno-based image reconstruction algorithm.

MATLAB-based time measurements of the new methods show a significant acceleration for all calculation steps compared to reference implementations with standard methods. E.g. for the Jacobian operations, improvement factors of well over 100 could be found.

The paper shows new methods for solving known computational tasks for solving inverse problems. A particular advantage is the coherent derivation and elaboration of the results. The approaches can also be applicable to other inverse problems.

The purpose of this paper is to reduce the artifacts in fast Bayesian reconstruction images in electrical tomography. This is in particular important with respect to object detection in electrical tomography applications.

The authors suggest to apply the Box-Cox transformation in Bayesian linear minimum mean square error (BMMSE) reconstruction to better accommodate the non-linear relation between the capacitance matrix and the permittivity distribution. The authors compare the results of the original algorithm with the modified algorithm and with the ground truth in both, simulation and experiments.

The results show a reduction of 50 percent of the mean square error caused by artifacts in low permittivity regions. Furthermore, the algorithm does not increase the computational complexity significantly such that the hard real time constraints can still be met. The authors demonstrate that the algorithm also works with limited observations angles. This allows for object detection in real time, e.g., in robot collision avoidance.

This paper shows that the extension of BMMSE by applying the Box-Cox transformation leads to a significant improvement of the quality of the reconstruction image while hard real time constraints are still met.