A Fuzzy PID-Controlled Iterative Calderon’s Method for Binary Distribution in Electrical Capacitance Tomography

Abstract

Electrical capacitance tomography (ECT) utilizes measured mutual capacitances across a region of interest to visualize distributions inside. As typical two-phase flows can be roughly treated as binary-valued material distributions, in this article, a fuzzy PID-controlled iterative algorithm is proposed for image reconstruction in cases of binary distributions. A closed-loop control system includes a fuzzy PID controller, Calderon’s method, and fast calculation of the Dirichlet-to-Neumann map. Capacitances measured in an electrode array of the ECT sensor are compared with the feedback, and the difference is input to the controller. Fuzzy rules are used to automatically adjust the three parameters of the controller, i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K_{P}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K_{I}$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K_{D}$ </tex-math></inline-formula> . The controller passes the difference to Calderon’s method for reconstructing permittivity distribution. Reconstructed distribution is used to calculate a boundary map for feedback, by fast calculation of the Dirichlet-to-Neumann map, and serves as an updated reference for measured capacitances. A smooth segmentation method is also introduced to deal with the binary distribution and release the fluctuation in the tuning of the PID controller. Numerical simulations were done to verify the performance of the proposed iterative Calderon’s method for binary distributions. Experiments on real phantoms were also carried out using an ECT system to evaluate the proposed method. Several distributions were set up with solid particles and air. The results show that the proposed method can produce images with clear edges and shapes of binary distributions.