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\(L^1\) regularization method in electrical impedance tomography by using the \(L^1\)-curve (Pareto frontier curve). (English) Zbl 1243.78041

Summary: Electrical impedance tomography (EIT), as an inverse problem, aims to calculate the internal conductivity distribution at the interior of an object from current-voltage measurements on its boundary. Many inverse problems are ill-posed, since the measurement data are limited and imperfect. To overcome ill-posedness in EIT, two main types of regularization techniques are widely used. One is categorized as the projection methods, such as truncated singular value decomposition (SVD or TSVD). The other categorized as penalty methods, such as Tikhonov regularization, and total variation methods. For both of these methods, a good regularization parameter should yield a fair balance between the perturbation error and regularized solution. In this paper a new method combining the least absolute shrinkage and selection operator (LASSO) and the basis pursuit denoising (BPDN) is introduced for EIT. For choosing the optimum regularization we use the \(L^1\)-curve (Pareto frontier curve) which is similar to the \(L\)-curve used in optimising \(L^2\)-norm problems. In the \(L^1\)-curve we use the \(L^1\)-norm of the solution instead of the \(L^2\) norm. The results are compared with the TSVD regularization method where the best regularization parameters are selected by observing the Picard condition and minimizing generalized cross validation (GCV) function. We show that this method yields a good regularization parameter corresponding to a regularized solution. Also, in situations where little is known about the noise level \(\sigma \), it is also useful to visualize the \(L^1\)-curve in order to understand the trade-offs between the norms of the residual and the solution. This method gives us a means to control the sparsity and filtering of the ill-posed EIT problem. Tracing this curve for the optimum solution can decrease the number of iterations by three times in comparison with using LASSO or BPDN separately.

MSC:

78A70 Biological applications of optics and electromagnetic theory
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
49N45 Inverse problems in optimal control
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