Shape Reconstruction Methods In Electrical Tomography

Electrical tomography is a non-invasive or non-intrusive process visualizationtechnology to image the internal electrical parameter (conductivity, permittivity orpermeability) distribution from the boundary measurements. Due to its advantages ofno radiation and low cost, it has been successfully applied in many industrialprocesses, such as: fluidized bed reaction, mixing process or multiphase flow.However, due to its non-linear and ill-posed character, its spatial resolution is low.By far, the most commonly used reconstruction algorithms in Electricaltomography are pixel/voxel based, in which the distribution of the electrical parameterare discretized into finite volume elements and then reconstructed by optimizing thefit of the simulated to measured boundary data. These methods are very suitable forimaging the mixed materials with indefinable interfaces, such as flames in porousmedia and commixture of organics. For the applications in immiscible materials withpiecewise constant distribution, such as flammable liquids detection and oiltransportation, it is difficult to obtain high spatial resolution images from thesemethods.To overcome this problem, some shape based reconstruction algorithms aredeveloped. These algorithms parameterize the distribution of the electrical parametersusing the boundaries of the piecewise constant subdomains. Correspondingly, theproblems of reconstructing the pixel/voxel images convert to the problem ofreconstructing the shapes of the boundaries of the subdomains. And a result, thedimensions of the reconstruction problem is reduced, and the ill-posed character ismitigated. The main woks in the paper accomplished as follows:1. The boundary element method is used to solve the forward problem inElectrical tomography. A concise boundary integral equation is derived by eliminatingthe independent variables in the system and adding some intermediate variables.2. A fast Jacobian calculation method is derived using the reciprocal theory andsome integral transformation technology. Comparing with the existing method, noadditional forward problems or linear systems are required to be solved in this method,and thus its computational efficiency is high. Furthermore the sensitivity field in the shape reconstruction problem is analyzed. Two rules are noticed. Firstly, the shapesensitivity is higher at the convex boundary than the concave boundary. Secondly, themeasurements are more sensitive to the shape changes near the electrodes.3. A shape reconstruction method is proposed to reconstruct the2D inclusionshapes based on the Fourier functions and the Levenberg-Marquardt algorithm. Theshape reconstruction method performed well both in the numerical and experimentaltests. And the results show that the reconstruction errors mainly come from theconcave parts of the inclusion boundaries.4. A shape reconstruction method is proposed to reconstruct the3D inclusionshapes based on the spherical harmonics and a multi-stage Levenberg-Marquardtstrategy. A3D ECT experiment platform is developed by using a cube ECT sensor.The results show that the proposed shape reconstruction method is effective in thereconstructing of the inclusion shapes. When the inclusion is near the electrodes, thereconstructed shapes are stretched toward the nearest electrodes.5. The2D interface is reconstructed using the Bezier function. A geometricalconstraint is proposed to restrain the interface on a meaningful shape. If theconductivity fluid can fill a quarter of the pipeline, the interfaces can be preciselyreconstructed from the ERT data using16electrodes.6. The3D interface is reconstructed using the Bezier surface. Since the Béziersurface representation provides a compact way to describe a3D interface using a fewparameters, search spaces of the inverse problem are reduced, and some smoothingconstraints on interface shape are implicitly introduced making the iteration morestable. To deal with the errors in the prior permittivity values, the permittivity andinterface shape are simultaneously reconstructed. The proposed method performedwell both in numerical and experimental tests.