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Some Regularization Methods For Electrical Impedance Tomography

Posted on:2016-09-15Degree:DoctorType:Dissertation
Country:ChinaCandidate:J WangFull Text:PDF
GTID:1108330479978618Subject:Mathematics
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Electrical Impedance Tomography( EIT) uses the conductivity distribution or variation inside the object as the imaging target, which attempts to estimate the internal electrical conductivity distribution from applied current and measured boundary voltages at surface electrodes. In comparison with the current existing imaging techniques, EIT is the new-emerging functional imaging and has the advantage of portable, inexpensive,non-invasive and so on. EIT problem is particularly attractive in many important applications and has excellent application prospect in medical imaging, geophysical exploration,industrial nondestructive testing, the search of underwater objects, etc. From the mathematical point of view, EIT imaging actually can be regarded as a kind of parameter identification inverse problem of second order elliptic partial differential equation. Since EIT inverse problem itself has some di?culties with nonlinearity, under-determination,severe ill-posedness, large amount of calculation and so on, which often lead to the low resolution and poor contrast for the reconstructed images, it is a challenging problem.The research of this dissertation is further focus on the image reconstruction algorithms for EIT inverse problem. The aim is to improve the resolution, robustness and real-time of the reconstructed images.Firstly, considering the linearized EIT model, for the inhomogeneous conductivity distribution which has sparsity in the spacial domain, we propose one sparsity constraint regularization model with l1-norm penalty term, also called l1-norm regularization. Since this regularization method involves the minimization of a kind of non-differential cost functional, we here introduce the fast and stable split Bregman iterative algorithm, which thus greatly simplifies the solving procedure for the original optimization problem and enhances the computational e?ciency. For this particular problem, numerical simulations with one single as well as multiple inclusions show that in contrast to the conventional l2-norm and TV regularization methods, l1-norm regularization method is of great advantage in eliminating the imaging artifacts in some degree and sharpening the edges of discontinuous inclusions in a constant background, and has well robustness against data noise. Moreover, the results also show the competitive performances of the split Bregman iterative algorithm in terms of the estimation of magnitude and edge location, as well as computational speed, in comparison with several existing l1-norm minimizing algorithms.Secondly, EIT inverse problem is typically nonlinear and the study on the nonlinear inversion model is very important. In an attempt to further improve the sharpness and robustness for the imaging problem, this work devises an elastic net regularization scheme with a convex combination of the smooth l2-norm prior and the sparsity-promoting l1-norm prior. This regularization scheme enhance both the l2-norm and l1-norm, and can simultaneously recover the smooth parts and edge regions of the small spatial profiles.The convergence and stability of the elastic net regularization scheme are discussed theoretically, and the convergence rate is obtained under proper conditions. We then provide one simple and fast alternating direction iteration optimization method based on the split Bregman iteration technique for solving the proposed objective function. Numerical simulations and tank experiments show that the proposed method with an appropriate choice for parameter will have well performance on sharpening the edges of the inclusions and is robust with respect to data noise. Moreover, in the tank experiments we test the effects on the reconstruction using different partition meshes for the forward and inverse solvers. It is founded that l2-norm regularization reconstruction becomes better with the finer meshes while the elastic net reconstruction is not affected notably with the change of partition meshes.Finally, based on the homotopy perturbation technique, we first use the homotopy theory to construct one homotopy equation containing an embedded parameter, and then introduce a novel updated iteration regularization scheme by using the principle of equality for the coe?cient of the embedding parameter. The convergence for the proposed updated iteration regularization scheme is discussed theoretically. As the improvement of the classical Landweber iteration method, this proposed scheme is simple and easy to be implemented, and belongs to the iterative regularization method, which does not involve selecting the regularization parameter. Simulations of image reconstruction have been performed to verify the feasibility and effectiveness. Numerical results indicate that this method can overcome the numerical instability and is robust to data noise in the EIT image reconstruction. Moreover, compared with the classical Landweber iteration method,our approach improves the convergence rate and reduces the computational time.
Keywords/Search Tags:electrical impedance tomography, sparsity constraint regularization method, split Bregman algorithm, elastic net regularization method, iterative regularization method
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