Comparison of linear and non-linear monotononicity-based shape reconstruction using exact matrix characterizations

Detecting inhomogeneities in the electrical conductivity is a special case of the inverse problem in electrical impedance tomography, that leads to fast direct reconstruction methods. One such method can, under reasonable assumptions, exactly characterize the inhomogeneities based on monotonicity properties of either the Neumann-to-Dirichlet map (non-linear) or its Fr\'echet derivative (linear). We give a comparison of the non-linear and linear approach in the presence of measurement noise, and show numerically that the two methods give essentially the same reconstruction in the unit disk domain. For a fair comparison, exact matrix characterizations are used when probing the monotonicity relations to avoid errors from numerical solution to PDEs and numerical integration. Using a special factorization of the Neumann-to-Dirichlet map also makes the non-linear method as fast as the linear method in the unit disk geometry.Comment: 18 pages, 5 figures, 1 tabl

Kun-dangwok: ''clan lects'' and Ausbau in western Arnhem Land

The sociolinguistic concept of an Ausbau language is widely thought of as exclusively associated with the standardization of languages for the political and social purposes of nation states. Language policy initiated by state institutions, the development of literacy and new specialist registers of language are typical elements involved in the Ausbau process. However, the linguistic ideologies of small language groups such as those of the minority languages of Aboriginal Australia can drive certain forms of deliberate language elaboration. An important aspect of Aboriginal linguistic ideology is language diversity, reflected in the development of elemental sociolinguistic varieties such as patrician lects. In the Bininj Kun-wok dialect chain of western Arnhem Land, a regional system of lectal differentiation known as Kun-dangwok has developed, reflecting an Aboriginal linguistic ideology whereby being different, especially different ways of speaking, are seen as central aspects of identity. The functions of the Kun-dangwok clan led system are described using examples of naturally occurring conversation which provide evidence that clan lects are the result of an Ausbau process that results in the opposite of language standardization and an increase in Abstand between varieties

Sparsity prior for electrical impedance tomography with partial data

This paper focuses on prior information for improved sparsity reconstruction in electrical impedance tomography with partial data, i.e. data measured only on subsets of the boundary. Sparsity is enforced using an $\ell_1$ norm of the basis coefficients as the penalty term in a Tikhonov functional, and prior information is incorporated by applying a spatially distributed regularization parameter. The resulting optimization problem allows great flexibility with respect to the choice of measurement boundaries and incorporation of prior knowledge. The problem is solved using a generalized conditional gradient method applying soft thresholding. Numerical examples show that the addition of prior information in the proposed algorithm gives vastly improved reconstructions even for the partial data problem. The method is in addition compared to a total variation approach.Comment: 17 pages, 12 figure

Convergence and regularization for monotonicity-based shape reconstruction in electrical impedance tomography

The inverse problem of electrical impedance tomography is severely ill-posed, meaning that, only limited information about the conductivity can in practice be recovered from boundary measurements of electric current and voltage. Recently it was shown that a simple monotonicity property of the related Neumann-to-Dirichlet map can be used to characterize shapes of inhomogeneities in a known background conductivity. In this paper we formulate a monotonicity-based shape reconstruction scheme that applies to approximative measurement models, and regularizes against noise and modelling error. We demonstrate that for admissible choices of regularization parameters the inhomogeneities are detected, and under reasonable assumptions, asymptotically exactly characterized. Moreover, we rigorously associate this result with the complete electrode model, and describe how a computationally cheap monotonicity-based reconstruction algorithm can be implemented. Numerical reconstructions from both simulated and real-life measurement data are presented

Distinguishability revisited: depth dependent bounds on reconstruction quality in electrical impedance tomography

The reconstruction problem in electrical impedance tomography is highly ill-posed, and it is often observed numerically that reconstructions have poor resolution far away from the measurement boundary but better resolution near the measurement boundary. The observation can be quantified by the concept of distinguishability of inclusions. This paper provides mathematically rigorous results supporting the intuition. Indeed, for a model problem lower and upper bounds on the distinguishability of an inclusion are derived in terms of the boundary data. These bounds depend explicitly on the distance of the inclusion to the boundary, i.e. the depth of the inclusion. The results are obtained for disk inclusions in a homogeneous background in the unit disk. The theoretical bounds are verified numerically using a novel, exact characterization of the forward map as a tridiagonal matrix.Comment: 25 pages, 6 figure