Approximation of full-boundary data from partial-boundary electrode measurements

Measurements on a subset of the boundary are common in electrical impedance tomography, especially any electrode model can be interpreted as a partial boundary problem. The information obtained is different to full-boundary measurements as modeled by the ideal continuum model. In this study we discuss an approach to approximate full-boundary data from partial-boundary measurements that is based on the knowledge of the involved projections. The approximate full-boundary data can then be obtained as the solution of a suitable optimization problem on the coefficients of the Neumann-to-Dirichlet map. By this procedure we are able to improve the reconstruction quality of continuum model based algorithms, in particular we present the effectiveness with a D-bar method. Reconstructions are presented for noisy simulated and real measurement data

Networks for Nonlinear Diffusion Problems in Imaging

A multitude of imaging and vision tasks have seen recently a major transformation by deep learning methods and in particular by the application of convolutional neural networks. These methods achieve impressive results, even for applications where it is not apparent that convolutions are suited to capture the underlying physics. In this work we develop a network architecture based on nonlinear diffusion processes, named DiffNet. By design, we obtain a nonlinear network architecture that is well suited for diffusion related problems in imaging. Furthermore, the performed updates are explicit, by which we obtain better interpretability and generalisability compared to classical convolutional neural network architectures. The performance of DiffNet tested on the inverse problem of nonlinear diffusion with the Perona-Malik filter on the STL-10 image dataset. We obtain competitive results to the established U-Net architecture, with a fraction of parameters and necessary training data

Deep D-Bar: Real-Time Electrical Impedance Tomography Imaging With Deep Neural Networks

The mathematical problem for electrical impedance tomography (EIT) is a highly nonlinear ill-posed inverse problem requiring carefully designed reconstruction procedures to ensure reliable image generation. D-bar methods are based on a rigorous mathematical analysis and provide robust direct reconstructions by using a low-pass filtering of the associated nonlinear Fourier data. Similarly to low-pass filtering of linear Fourier data, only using low frequencies in the image recovery process results in blurred images lacking sharp features, such as clear organ boundaries. Convolutional neural networks provide a powerful framework for post-processing such convolved direct reconstructions. In this paper, we demonstrate that these CNN techniques lead to sharp and reliable reconstructions even for the highly nonlinear inverse problem of EIT. The network is trained on data sets of simulated examples and then applied to experimental data without the need to perform an additional transfer training. Results for absolute EIT images are presented using experimental EIT data from the ACT4 and KIT4 EIT systems

A Data-Driven Edge-Preserving D-bar Method for Electrical Impedance Tomography

In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known \emph{a priori} that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called {\em CGO sinogram}. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.Comment: 24 pages, 11 figure

Advances in D-bar methods for partial boundary data electrical impedance tomography : From continuum to electrode models and back

Electrical impedance tomography (EIT) is a rather new approach to medical imaging that is motivated by using electricity to determine the inside of a body. The clear advantage lies in the usage of harmless electric currents, in contrast to the ionizing radiation of X-rays, whereas the mathematical problem is inherently more challenging. In EIT we seek to reconstruct an image of the inner organs by determining their conductivity, i.e. how well electricity is conducted. As a medical imaging modality it is most promising in pulmonary and cardiac imaging, due to considerably different conductivity values in the air filled lungs (low conductive) and the blood filled heart (high conductive). EIT is in principle capable of monitoring the respiratory process, detecting pathologies in the lungs, and monitoring the heart activity. The main focus of this work is on the partial-boundary problem in EIT, that means one has only access to a certain part of the boundary and data can only be collected there. In a hospital setting these situations can arise when monitoring a critical or unconscious patient and hence one can only access the front of the torso (ventral position). Furthermore, practical complications can arise due to faulty, dislocated, or dispatched electrodes and hence leading to incomplete data. The methods presented in this thesis are capable of dealing with such incomplete data. Following the tradition of mathematical research we are also interested in quantifying the error incomplete data introduces to the reconstruction. In a short summary, this thesis investigates how to improve EIT reconstructions from partial-boundary data by utilizing concepts from an ideal mathematical setting as well as how to apply these methods to real electrode models and measurement data.Sähköinen impedanssitomografia (engl. electrical impedance tomography, lyh. EIT) on kohtalaisen uusi lääketieteellisen kuvantamisen muoto, jossa potilaan sisälle pyritään näkemään heikon sähkövirran avulla. Verrattuna ionisoivaan säteilyyn, kuten röntgensäteilyyn, EIT-kuvantaminen on potilaalle vähemmän haitallista. Kääntöpuolena on tähän kuvantamismuotoon liittyvän matematiikan haastavuus. EIT-kuvantamisessa tavoitteena on tutkia potilaan sisäelimiä selvittämällä niiden sähkönjohtavuus, eli se kuinka hyvin sisäelimet johtavat sähköä. Lääketieteellisen kuvantamisen muodoista se soveltuu erityisen hyvin sydämen ja keuhkojen kuvantamiseen. Tämä johtuu siitä, että sisään hengittäessä keuhkoissa on paljon ilmaa ja toisaalta sydämessä paljon verta. Siksi keuhkojen sähkönjohtavuus on huomattavasti sydäntä pienempi. EIT:llä voidaan tarkkailla potilaan hengitystä, keuhkojen fyysisiä vikoja sekä sydämen aktiivisuutta. Tämän työn pääteemana on EIT-kuvantamisen osittaiset reuna-arvo-ongelmat. Tämä tarkoittaa sitä, että mittauksia tehdään vain osassa mitattavan kappaleen ulkopintaa. Sairaalaolosuhteissa eräs esimerkki tällaisesta mittaustilanteesta on kriittisessä tai tajuttomassa tilassa olevan potilaan tutkiminen, jolloin mittaavia elektroneita voidaan asentaa potilaalle ainoastaan rinnan alueelle. Tämän lisäksi EIT-mittauksessa puutteellista mittausdataa saattavat aiheuttaa käytännön mittausvirheet, jotka voivat johtua viallisista, väärin asennetutuista tai väärässä paikassa olevista elektroneista. Tässä työssä esitetyt menetelmät tarjoavat matemaattisia sekä käytännön mittaustilanteissa tarvittavia keinoja edellä kuvattujen ongelmien ratkaisemiseksi. Matemaattista traditiota noudattaen olemme myös kiinnostuneita kvantifioimaan puutteellisen datan avulla tehtyjen rekonstruktioiden virheellisyyttä. Tiivistäen, tässä työssä tutkitaan kuinka parantaa EIT-kuvantamisen avulla tehtyjä rekonstruktioita, kun vaillinaiset mittaukset on tehty vain osassa kappaleen ulkopintaa. Ratkaisemme matemaattisen ongelman ja sovellamme tämän ongelman ratkaisua todellisen maailman mittausongelmiin

A Direct D-Bar Method for Partial Boundary Data Electrical Impedance Tomography With a Priori Information

Electrical Impedance Tomography (EIT) is a non-invasive imaging modality that uses surface electrical measurements to determine the internal conductivity of a body. The mathematical formulation of the EIT problem is a nonlinear and severely ill-posed inverse problem for which direct D-bar methods have proved useful in providing noise-robust conductivity reconstructions. Recent advances in D-bar methods allow for conductivity reconstructions using EIT measurement data from only part of the domain (e.g., a patient lying on their back could be imaged using only data gathered on the accessible part of the body). However, D-bar reconstructions suffer from a loss of sharp edges due to a nonlinear low-pass filtering of the measured data, and this problem becomes especially marked in the case of partial boundary data. Including a priori data directly into the D-bar solution method greatly enhances the spatial resolution, allowing for detection of underlying pathologies or defects, even with no assumption of their presence in the prior. This work combines partial data D-bar with a priori data, allowing for noise-robust conductivity reconstructions with greatly improved spatial resolution. The method is demonstrated to be effective on noisy simulated EIT measurement data simulating both medical and industrial imaging scenarios

Essays on Wage Formation and Globalization

Why do some firms pay collectively agreed wages rather than to negotiate wages individually? Do exporting firms pay higher wages than non-exporting firms, and to what extent this is determined by institutional frameworks? What are the connections between the labor unit costs and a strong export performance of companies? These and other questions are addressed in this book. In several chapters the author shows a variety of interactions between wages, globalization and institutional factors.Warum zahlen manche Firmen nach Tarif, statt die Löhne individuell auszuhandeln? Zahlen Exportfirmen höhere Löhne als Firmen, die nicht exportieren und inwieweit wird dies von institutionellen Rahmenbedingungen bestimmt? Welche Zusammenhänge bestehen zwischen den Lohnstückkosten und der Exportstärke von Unternehmen? Mit diesen und weiteren Fragen befasst sich der Autor im vorliegenden Band. In mehreren Kapiteln legt er dar, dass zwischen Löhnen, Globalisierung und institutionellen Kontextfaktoren vielfältige Wechselwirkungen bestehen