Quantitative Study On Influence Of Finite Element Approximation Error On Eit Forward And Inverse Problem And Application Of Operator Factorization Method For EIT

Electrical impedance tomography (EIT) is mathematically a forward andinverse problem described by elliptic partial differential equation. The forwardproblem calculates the inner field distribution according to the model geometryand source parameters. The inverse problem reconstructs the field distributionbased on the measured boundary voltages. The research of forward problemincludes mathematical physics theory, mathematical model and the solution oflarge-scale equations. The inverse problem is always ill-posed. That is, evenvery small errors in the forward solution will have a great impact onreconstruction. In other words, reconstruction is highly sensitive to measurementand modeling errors. Measurement errors have been studied more widely thanmodeling errors and it has been common practice to ignore modeling errors. Themodeling errors are considered to be the truncation of the computation domain,unknown electrode contact impedances and unknown shape of the measurementtarget. These errors are pivotal in EIT, since they determine the reconstructionefficiency and quality of image. The finite element method (FEM)approximation error belongs to the general problems in engineering and mathematics. So, our numerical simulation study focuses on their quantitativeeffects on EIT forward and inverse problem.Typically, the FEM approximation error depends on mesh density and theorder of the interpolation function. The main work which is on the basis of alarge amount of numerical experiments is described as follows:(1) Error Analysis for EIT forward problemThe calculation precision, the error, the convergent order and computationtime between triangular and quadrilateral elements were studied using aparticular case under different mesh density and order of interpolation function.The numerical experiments indicated that the accuracy became higher as thenumber of elements and the order of interpolation function increased, along withthe increase of calculation consumption. With the increase of the order, theaccuracy increased non-linearly.We set up the analytical solution formula for the uniform circular andthree-layer concentric circular models. We quantified the accuracy using variousnumerical analysis indicators. The results indicated that the enlargement of themeshing scale played a limited role in improving the calculation accuracy for acertain element of high order, such as second or third order. Furthermore, theimprovement of the higher order function is poor. Specifically, the errors amongthe second、third and four order elements only varied with a0.01order ofmagnitude.(2) The influence on the reconstruction of2D and3D imagesThe NOSER and TV regularization algorithms were used in simulation tostudy the influence of approximation error. We put forward three objective andeffective evaluation indicators: Reconstruction Quality Function D, StructuralSimilarity Image Measurement SSIM and Total Error TE. Reconstruction quality became better as the elements increased. Three indicators showed thatthe image improved after Bayesian approximation error approach was adopted.The above properties remained the same with the presence of noise of varioussignal noise ratio, although there were many artifacts. Therefore the mediumresolution was sufficient for linear noniterative algorithm such as NOSER.The value of TE decreased with the increase of iteration in the lowresolution mesh, whereas the error increased and convergence rate slowed downor the algorithm itself became non-convergent in the high resolution mesh.Hence the low resolution mesh with adequate iteration was good for iterativealgorithms with TV regularization.In three dimension complete electrode model, the boundary data error wasless as the number of elements and nodes increased. Seven different mesheswith increasing meshing scale were used in the forward model while mediumresolution mesh was employed in the inverse model. Results with better quality,less artifact and sharper boundary were obtained.(3) Application of the operator factorization method for EITThe operator factorization method is a new tool for direct reconstruction inEIT when the Neumann-Dirichlet operator is known. We sketched the basicproperties and the main results of the mathematical model with regularization.Satisfied results for multi-target model were obtained in simulation.In conclusion, numerical analysis was conducted based on substantialsimulations. Quantitive evaluation of influence of FEM meshing andinterpolation function of different orders on forward problem was carried out.Error analysis on numerical solution using analytical formula of uniform regionwas realized. The impact of FEM meshing scale on image reconstruction wasdiscussed quantitively. Finally, the operator factorization method that belongs to the direct method, one of the three fundamental approaches to solve EIT inverseproblems, was presented. This lays the preliminary theoretical basis and erroranalysis for more comprehensive evaluation of EIT in the future.