Research On The Boundary Reconstruction Problem Of Electrical Resistance Tomography

As a kind of electrical tomography techniques, electrical resistance tomography (ERT) is a novel technique to measure cross-sectional conductivity distribution, which has been developed since late 1980s. ERT has advantages in cost, speed and simplicity, and possesses a non-intrusive nature, thus it receives much concern in many research fields. However, the inverse conductivity problem is severely non-linear and ill-posed, which makes difficulties to get a precise and stable reconstruction. The widely acceptable idea to handle problems of this kind is to take full advantage of prior information and set least unknowns in the inverse problem solver. The conductivity reconstruction problem of ERT is transformed to determination of the boundary of inclusions in this paper, where the amount of unknowns is sharply decreased and the constant conductivity of objects is regarded as prior information.The presented boundary reconstruction method is quantitative, thus the geometric and distributional information of objects can be easily achieved from the result. Main efforts and results are as follows.Firstly, the boundary integral equations of ERT sensing field under complete electrode model are deduced, where the non-independent variables are eliminated and a set of linear equations with minimum variables is obtained. Compared to the existing studies, the amount of variables has been decreased at a theoretic maximum extent of 50% variables. The analytical integrations for quadratic elements are presented so that higher-order integrals are expressed by lower-order ones for rapid calculation, which ensures the precision of the ERT forward problem solver.Secondly, a direct linearizing method is presented to calculate the Jacobian matrix based on the analytic boundary integrations. The arrangement of the system matrix and the utilization of QR decomposition ensure rapid calculation of Jacobian matrix, and analytic integrations guarantee the precision.Thirdly, a boundary reconstruction algorithm is presented based on Levenberg-Marquardt method. Much importance is put on the smoothness of boundaries in the reconstruction, thus a restriction of the curve radius is introduced to adjust the damping parameter. Analysis results on the stability and precision of the boundary reconstruction are given. Stable results can be achieved when the conductivity of the objects differs much from that of background medium, and results for convex boundaries are precise. Reversely, the reconstructions for inclusions with similar conductivities to the background medium are not stable. The ability of the modified boundary reconstruction algorithm to approximate the true curve is discussed when the prior information is wrongly given.Fourthly, new integral equations are deduced for cases where complete conducting inclusions are encountered. Then the boundary reconstruction is also achieved.Finally, a one-step boundary reconstruction method is presented to obtain the separated interface of horizontal gas/liquid two phase flows, based on a rough estimation of the liquid level from the boundary measurements.