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

Comparison of total variation algorithms for electrical impedance tomography

The applications of total variation (TV) algorithms for electrical impedance tomography (EIT) have been investigated. The use of the TV regularisation technique helps to preserve discontinuities in reconstruction, such as the boundaries of perturbations and sharp changes in conductivity, which are unintentionally smoothed by traditional l2 norm regularisation. However, the non-differentiability of TV regularisation has led to the use of different algorithms. Recent advances in TV algorithms such as the primal dual interior point method (PDIPM), the linearised alternating direction method of multipliers (LADMM) and the spilt Bregman (SB) method have all been demonstrated successful EIT applications, but no direct comparison of the techniques has been made. Their noise performance, spatial resolution and convergence rate applied to time difference EIT were studied in simulations on 2D cylindrical meshes with different noise levels, 2D cylindrical tank and 3D anatomically head-shaped phantoms containing vegetable material with complex conductivity. LADMM had the fastest calculation speed but worst resolution due to the exclusion of the second-derivative; PDIPM reconstructed the sharpest change in conductivity but with lower contrast than SB; SB had a faster convergence rate than PDIPM and the lowest image errors

Transformer Meets Boundary Value Inverse Problems

A Transformer-based deep direct sampling method is proposed for a class of boundary value inverse problems. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural networks? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data in different frequencies are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions as different input channels. Then, by introducing learnable non-local kernels, the direct sampling is recast to a modified attention mechanism. The proposed method is then applied to electrical impedance tomography, a well-known severely ill-posed nonlinear inverse problem. The new method achieves superior accuracy over its predecessors and contemporary operator learners, as well as shows robustness with respect to noise. This research shall strengthen the insights that the attention mechanism, despite being invented for natural language processing tasks, offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures

Tecniche Elettrotomografiche per la caratterizzazione dei tessuti biologici

Electrical impedance tomography (EIT) is an imaging modality wherein the spatial map of conductivity and permittivity inside a medium is obtained from a set of surface electrical measurements. Electrodes are brought into contact with the surface of the object being imaged and a set of currents are applied and the corresponding voltages are measured. These voltages and currents are then used to estimate the electrical properties of the object using an image reconstruction algorithm which relies on an accurate model of the electrical interaction. The process of property estimation, called inverse problem, is highly ill-posed and it requires a Regularization method. The objective of this Thesis was to develop a device for imaging using the EIT technique, which was convenient, noninvasive, easily programmable, portable and relatively cheap in contrast to many other diagnostic tool. In this direction a simple EIT system and its hardware and software parts are developed. The data processing was accomplished by utilizing the EIDORS toolkit, which was developed for application to this nonlinear and ill-posed inverse problem. Experiments have indicated that the EIT system can reconstruct resistive and capacitive images of good contrast despite errors in the measurement are not taken in account

Robust Computation in 2D Absolute EIT (A-EIT) Using D-Bar Methods with the “EXP” Approximation

Objective Absolute images have important applications in medical Electrical Impedance Tomography (EIT) imaging, but the traditional minimization and statistical based computations are very sensitive to modeling errors and noise. In this paper, it is demonstrated that D-bar reconstruction methods for absolute EIT are robust to such errors. Approach The effects of errors in domain shape and electrode placement on absolute images computed with 2-D D-bar reconstruction algorithms are studied on experimental data. Main Results It is demonstrated with tank data from several EIT systems that these methods are quite robust to such modeling errors, and furthermore the artefacts arising from such modeling errors are similar to those occurring in classic time-difference EIT imaging. Significance This study is promising for clinical applications where absolute EIT images are desirable, but previously thought impossible

Monotoniemethoden für inverse Parameteridentifikationsprobleme partieller Differentialgleichungen

This work is concerned with monotonicity-based methods for inverse parameter reconstruction problems in partial differential equations. The first three chapters address the anomaly detection problem of electrical impedance tomography. While electrical impedance tomography aims on reconstructing the interior conductivity distribution of a conductive subject from boundary data, the goal of the specific anomaly detection problem is the reconstruction of areas inside a conductive subject where the conductivity differs from an expected reference conductivity. The considered boundary data can be understood as an operator that describes current-voltage measurements. In the final chapter we prove a novel uniqueness result for the inverse potential problem of the Schrödinger equation with partial data. For the development of anomaly detection methods, both known and novel variants of a monotonicity relation are used. Roughly speaking, these monotonicity relations particularly show that a pointwise decrease of the conductivity leads to larger boundary data (in sense of operator definiteness). At first glance, it is not obvious at all whether the converse of this implication holds also true, i.e., it is not clear whether larger boundary data could also result from a local decrease of the conductivity in some parts and a local increase in other parts. Assuming a local definiteness condition for the conductivity change we prove a partial converse of the monotonicity implication that holds for the case in which the measurements are modeled with the idealized continuum model. In the first chapter we develop novel anomaly detection methods for measurement data modeled with the continuum model. Moreover, fast linearized variants are presented that only require the computation of reference measurements for one homogeneous reference conductivity. We prove that all presented methods are capable of reconstructing the exact outer shape of conductivity anomalies. In realistic electrical impedance tomography settings in which measurement data is collected on a finite number of electrodes, the reconstruction of the exact outer shape of anomalies cannot be guaranteed anymore. On top of that, systematic errors resulting from imprecise knowledge of the setting parameters as well as additional random measurement errors need to be taken into account. In the second chapter we show that nevertheless certain resolution guarantees are principally possible for such settings. In the third chapter we develop a novel hybrid method that does not require the simulation of reference data. We apply an idealized model for ultrasound modulation that alters the conductivity uniformly in a test region and we develop a test criterion to check whether the test region is located inside an anomaly. The test criterion consists of a monotonicity-based comparison of ultrasound modulated and weighted frequency-difference measurements. Finally, in the fourth chapter, a local uniqueness result for the inverse potential problem of the Schrödinger equation on a bounded Lipschitz domain with partial boundary data is shown. More precisely, we show that positive-valued bounded potentials that do not completely coincide in a neighborhood of a potentially arbitrarily small part of the boundary can be distinguished from Cauchy data on this boundary part provided that a local definiteness condition is fulfilled.Die vorliegende Arbeit behandelt Monotoniemethoden für inverse Parameteridentifikationsprobleme partieller Differentialgleichungen. Die ersten drei Kapitel befassen sich mit dem Detektionsproblem von Leitfähigkeitsanomalien der elektrischen Impedanztomographie. Während die elektrische Impedanztomographie die Rekonstruktion der inneren Leitfähigkeitsverteilung eines leitenden Subjekts aus Randdaten zum Ziel hat, geht es bei dem speziellen Anomaliedetektionsproblem um die Rekonstruktion von Gebieten innerhalb eines leitenden Subjekts in denen die Leitfähigkeit von einer erwarteten Referenzleitfähigkeit abweicht. Die betrachteten Randdaten können dabei als ein Operator verstanden werden, welcher Strom-zu-Spannungsmessungen beschreibt. Im finalen Kapitel beweisen wir ein neues Eindeutigkeitsresultat für das inverse Potentialproblem der Schrödingergleichung mit partiellen Randdaten. Für die Entwicklung von Methoden zur Anomaliedetektion verwenden wir sowohl bekannte als auch neue Varianten einer Monotonierelation. Anschaulich formuliert, besagen diese Monotonierelationen insbesondere, dass eine punktweise Verringerung der Leitfähigkeit zu größeren Randdaten (im Sinne einer Definitheitsrelation für Operatoren) führt. Auf den ersten Blick ist überhaupt nicht ersichtlich, ob die umgekehrte Implikation ebenfalls gilt. Das heißt, es ist nicht klar, ob größere Randdaten auch aus einer lokalen Verringerung der Leitfähigkeit in einem Bereich und einer lokalen Erhöhung in einem anderen Bereich resultieren könnten. Unter Voraussetzung einer lokale Definitheitsbedingung für die Leitfähigkeitsänderung beweisen wir eine partielle Umkehrung zur Implikation der Monotonierelation, die für Randdaten in einem idealisierten Setting (Continuum Model) gilt. Im ersten Kapitel entwickeln wir neuartige Monotoniemethoden zur Anomaliedetektion für Randdaten modelliert mit dem Continuum Model. Zudem präsentieren wir schnelle linearisierte Varianten, welche nur die Berechnung von Referenzdaten für eine einzige homogene Referenzleitfähigkeit benötigen. Wir beweisen, dass alle präsentierten Methoden die exakte äußere Form von Anomalien rekonstruieren. In einem realistischen Setting, in dem Messdaten auf einer endlichen Anzahl von Elektroden gesammelt werden, lässt sich die Rekonstruktion der exakten äußeren Form von Anomalien nicht mehr garantieren. Erschwerend kommt hinzu, dass systematische Fehler, resultierend aus ungenauer Kenntnis von Settingparametern, sowie zufällige Messfehler berücksichtigt werden müssen. Im zweiten Kapitel zeigen wir, dass gewisse Auflösungsgarantien dennoch auch für solche Settings prinzipiell möglich sind. Im dritten Kapitel entwickeln wir eine neuartige Hybridmethode, die ohne die Simulation von Referenzdaten auskommt. Wir verwenden ein idealisiertes Modell zur Ultraschallmodulation, das die Leitfähigkeit gleichmäßig in einer Testregion erhöht und entwickeln ein Testkriterium, um zu testen, ob die Testregion innerhalb einer Anomalie liegt. Das Testkriterium besteht in einem monotoniebasierten Vergleich von Ultraschall-modulierten und gewichteten Frequenz-Differenz-Messungen. Abschließend wird im vierten Kapitel ein lokales Eindeutigkeitsresultat für das inverse Potentialproblem der Schrödingergleichung auf einem beschränkten Lipschitz-Gebiet für partielle Randdaten bewiesen. Genauer gesagt zeigen wir, dass sich positivwertige beschränkte Potentiale, die in einer Umgebung eines Randstücks nicht komplett übereinstimmen, anhand von Cauchy-Daten auf diesem Randstück unterscheiden lassen, vorausgesetzt eine lokale Definitheitsbedingung ist erfüllt