Die Faktorisierungsmethode für die elektrische Impedanztomographie im Halbraum

In der vorliegenden Arbeit wird die Faktorisierungsmethode zur Erkennung von Inhomogenitäten der Leitfähigkeit in der elektrischen Impedanztomographie auf unbeschränkten Gebieten - speziell der Halbebene bzw. dem Halbraum - untersucht. Als Lösungsräume für das direkte Problem, d.h. die Bestimmung des elektrischen Potentials zu vorgegebener Leitfähigkeit und zu vorgegebenem Randstrom, führen wir gewichtete Sobolev-Räume ein. In diesen wird die Existenz von schwachen Lösungen des direkten Problems gezeigt und die Gültigkeit einer Integraldarstellung für die Lösung der Laplace-Gleichung, die man bei homogener Leitfähigkeit erhält, bewiesen. Mittels der Faktorisierungsmethode geben wir eine explizite Charakterisierung von Einschlüssen an, die gegenüber dem Hintergrund eine sprunghaft erhöhte oder erniedrigte Leitfähigkeit haben. Damit ist zugleich für diese Klasse von Leitfähigkeiten die eindeutige Rekonstruierbarkeit der Einschlüsse bei Kenntnis der lokalen Neumann-Dirichlet-Abbildung gezeigt. Die mittels der Faktorisierungsmethode erhaltene Charakterisierung der Einschlüsse haben wir in ein numerisches Verfahren umgesetzt und sowohl im zwei- als auch im dreidimensionalen Fall mit simulierten, teilweise gestörten Daten getestet. Im Gegensatz zu anderen bekannten Rekonstruktionsverfahren benötigt das hier vorgestellte keine Vorabinformation über Anzahl und Form der Einschlüsse und hat als nicht-iteratives Verfahren einen vergleichsweise geringen Rechenaufwand.

Parallel modeling of the electric field distribution in the brain

The term "Brain Imaging" identi�es a set of techniques to analyze the structure and/or functional behavior of the brain in normal and/or pathological situations. These techniques are largely used in the study of brain activity. In addition to clinical usage, analysis of brain activity is gaining popularity in others recent �fields, i.e. Brain Computer Interfaces (BCI) and the study of cognitive processes. In this context, usage of classical solutions (e.g. f MRI, PET-CT) could be unfeasible, due to their low temporal resolution, high cost and limited portability. For these reasons alternative low cost techniques are object of research, typically based on simple recording hardware and on intensive data elaboration process. Typical examples are ElectroEncephaloGraphy (EEG) and Electrical Impedance Tomography (EIT), where electric potential at the patient's scalp is recorded by high impedance electrodes. In EEG potentials are directly generated from neuronal activity, while in EIT by the injection of small currents at the scalp. To retrieve meaningful insights on brain activity from measurements, EIT and EEG relies on detailed knowledge of the underlying electrical properties of the body. This is obtained from numerical models of the electric �field distribution therein. The inhomogeneous and anisotropic electric properties of human tissues make accurate modeling and simulation very challenging, leading to a tradeo�ff between physical accuracy and technical feasibility, which currently severely limits the capabilities of these techniques. Moreover elaboration of data recorded requires usage of regularization techniques computationally intensive, which influences the application with heavy temporal constraints (such as BCI). This work focuses on the parallel implementation of a work-flow for EEG and EIT data processing. The resulting software is accelerated using multi-core GPUs, in order to provide solution in reasonable times and address requirements of real-time BCI systems, without over-simplifying the complexity and accuracy of the head models.

Heterogeneous multicore systems for signal processing

This thesis explores the capabilities of heterogeneous multi-core systems, based on multiple Graphics Processing Units (GPUs) in a standard desktop framework. Multi-GPU accelerated desk side computers are an appealing alternative to other high performance computing (HPC) systems: being composed of commodity hardware components fabricated in large quantities, their price-performance ratio is unparalleled in the world of high performance computing. Essentially bringing “supercomputing to the masses”, this opens up new possibilities for application fields where investing in HPC resources had been considered unfeasible before. One of these is the field of bioelectrical imaging, a class of medical imaging technologies that occupy a low-cost niche next to million-dollar systems like functional Magnetic Resonance Imaging (fMRI). In the scope of this work, several computational challenges encountered in bioelectrical imaging are tackled with this new kind of computing resource, striving to help these methods approach their true potential. Specifically, the following main contributions were made: Firstly, a novel dual-GPU implementation of parallel triangular matrix inversion (TMI) is presented, addressing an crucial kernel in computation of multi-mesh head models of encephalographic (EEG) source localization. This includes not only a highly efficient implementation of the routine itself achieving excellent speedups versus an optimized CPU implementation, but also a novel GPU-friendly compressed storage scheme for triangular matrices. Secondly, a scalable multi-GPU solver for non-hermitian linear systems was implemented. It is integrated into a simulation environment for electrical impedance tomography (EIT) that requires frequent solution of complex systems with millions of unknowns, a task that this solution can perform within seconds. In terms of computational throughput, it outperforms not only an highly optimized multi-CPU reference, but related GPU-based work as well. Finally, a GPU-accelerated graphical EEG real-time source localization software was implemented. Thanks to acceleration, it can meet real-time requirements in unpreceeded anatomical detail running more complex localization algorithms. Additionally, a novel implementation to extract anatomical priors from static Magnetic Resonance (MR) scansions has been included.

Entwicklung und Tests von Algorithmen für die Impedanztomographie

Die Elektrische Impedanztomographie soll als kostengünstige und nebenwirkungsfreie Tomographiemethode in der medizinischen Diagnostik, z. B. in der Mammographie dienen. Mit der EIT läßt sich Krebsgewebe von gesundem Gewebe unterscheiden, da es eine signifikant erhöhte Leitfähigkeit aufweist. Damit kann die EIT als Ergänzung zu den klassischen Diagnoseverfahren dienen. So ist z.B. bei jungen Frauen mit einem dichteren Fettgewebe die Identifizierung eines Mammakarzinoms mit der Röntgentomographie nicht immer möglich. Ziel dieser Arbeit war es, einen Prototypen für die Impedanztomographie zu entwickeln und mögliche Anwendungen zu testen. Der Tomograph ist in Zusammenarbeit mit Dr. K.H.Georgi gebaut worden. Der Tomograph erlaubt es niederohmige, Wechselströme an Elektroden auf der Körperoberfläche einzuspeisen. Die Potentiale können an diesen Elektroden programmierbar vorgegeben werden. Weitere hochohmige Elektroden dienen zur Potentialmessung. Um den Hautwiderstand zu überbrücken, werden Wechselstromfrequenzen von 20-100 kHz eingesetzt. Mit der Möglichkeit der Messung von Strom und Potential auf unterschiedlichen Elektroden kann man das Problem des nur ungenau bekannten Hautwiderstandes umgehen. Prinzipiell ist es mit dem Mainzer EIT System möglich, 100 Messungen in der Sekunde durchzuführen. Auf der Basis von mit dem Mainzer EIT gewonnenen Daten sollten unterschiedliche Rekonstruktionsalgorithmen getestet und weiterentwickelt werden. In der Vergangenheit sind verschiedene Rekonstruktionsalgorithmen für das mathematisch schlecht gestellte EIT Problem betrachtet worden. Sie beruhen im Wesentlichen auf zwei Strategien: Die Linearisierung und iterative Lösung des Problems und Gebietserkennungsmethoden. Die iterativen Verfahren wurden von mir dahingehend modifiziert, dass Leitfähigkeitserhöhungen und Leitfähigkeitserniedrigungen gleichberechtigt behandelt werden können. Für den modifizierten Algorithmus wurden zwei verschiedene Rekonstruktionsalgorithmen programmiert und mit synthetischen Daten getestet. Zum einen die Rekonstruktion über die approximative Inverse, zum anderen eine Rekonstruktion mit einer Diskretisierung. Speziell für die Rekonstruktion mittels Diskretisierung wurde eine Methode entwickelt, mit der zusätzliche Informationen in der Rekonstruktion berücksichtigt werden können, was zu einer Verbesserung der Rekonstruktion beiträgt. Der Gebietserkennungsalgorithmus kann diese Zusatzinformationen liefern. In der Arbeit wurde ein neueres Verfahren für die Gebietserkennung derart modifiziert, dass eine Rekonstruktion auch für getrennte Strom- und Spannungselektroden möglich wurde. Mit Hilfe von Differenzdaten lassen sich ausgezeichnete Rekonstruktionen erzielen. Für die medizinischen Anwendungen sind aber Absolutmessungen nötig, d.h. ohne Leermessung. Der erwartende Effekt einer Inhomogenität in der Leitfähigkeit ist sehr klein und als Differenz zweier grosser Zahlen sehr schwierig zu bestimmen. Die entwickelten Algorithmen kommen auch gut mit Absolutdaten zurecht.

Analog Signal Acquisition and Conditioning for Near-Field Capacitive Communication and Active Combined EEG-EIT Monitoring

The work of the present thesis is focused on the implementation of microelectronic voltage sensing devices, with the purpose of transmitting and extracting analog information between devices of different nature at short distances or upon contact. Initally, chip-to-chip communication has been studied, and circuitry for 3D capacitive coupling has been implemented. Such circuits allow the communication between dies fabricated in different technologies. Due to their novelty, they are not standardized and currently not supported by standard CAD tools. In order to overcome such burden, a novel approach for the characterization of such communicating links has been proposed. This results in shorter design times and increased accuracy. Communication between an integrated circuit (IC) and a probe card has been extensively studied as well. Today wafer probing is a costly test procedure with many drawbacks, which could be overcome by a different communication approach such as capacitive coupling. For this reason wireless wafer probing has been investigated as an alternative approach to standard on-contact wafer probing. Interfaces between integrated circuits and biological systems have also been investigated. Active electrodes for simultaneous electroencephalography (EEG) and electrical impedance tomography (EIT) have been implemented for the first time in a 0.35 um process. Number of wires has been minimized by sharing the analog outputs and supply on a single wire, thus implementing electrodes that require only 4 wires for their operation. Minimization of wires reduces the cable weight and thus limits the patient's discomfort. The physical channel for communication between an IC and a biological medium is represented by the electrode itself. As this is a very crucial point for biopotential acquisitions, large efforts have been carried in order to investigate the different electrode technologies and geometries and an electromagnetic model is presented in order to characterize the properties of the electrode to skin interface.

Ein inverses Rückstreuproblem der elektrischen Impedanztomographie

Die vorliegende Arbeit untersucht das inverse Hindernisproblem der zweidimensionalen elektrischen Impedanztomographie (EIT) mit Rückstreudaten. Wir präsentieren und analysieren das mathematische Modell für Rückstreudaten, diskutieren das inverse Problem für einen einzelnen isolierenden oder perfekt leitenden Einschluss und stellen zwei Rekonstruktionsverfahren für das inverse Hindernisproblem mit Rückstreudaten vor. Ziel des inversen Hindernisproblems der EIT ist es, Inhomogenitäten (sogenannte Einschlüsse) der elektrischen Leitfähigkeit eines Körpers aus Strom-Spannungs-Messungen an der Körperoberfläche zu identifizieren. Für die Messung von Rückstreudaten ist dafür nur ein Paar aus an der Körperoberfläche nahe zueinander angebrachten Elektroden nötig, das zur Datenerfassung auf der Oberfläche entlang bewegt wird. Wir stellen ein mathematisches Modell für Rückstreudaten vor und zeigen, dass Rückstreudaten die Randwerte einer außerhalb der Einschlüsse holomorphen Funktion sind. Auf dieser Grundlage entwickeln wir das Konzept des konvexen Rückstreuträgers: Der konvexe Rückstreuträger ist eine Teilmenge der konvexen Hülle der Einschlüsse und kann daher zu deren Auffindung dienen. Wir stellen einen Algorithmus zur Berechnung des konvexen Rückstreuträgers vor und demonstrieren ihn an numerischen Beispielen. Ferner zeigen wir, dass ein einzelner isolierender Einschluss anhand seiner Rückstreudaten eindeutig identifizierbar ist. Der Beweis dazu beruht auf dem Riemann'schen Abbildungssatz für zweifach zusammenhängende Gebiete und dient als Grundlage für einen Rekonstruktionsalgorithmus, dessen Leistungsfähigkeit wir an verschiedenen Beispielen demonstrieren. Ein perfekt leitender Einschluss ist hingegen nicht immer aus seinen Rückstreudaten rekonstruierbar. Wir diskutieren, in welchen Fällen die eindeutige Identifizierung fehlschlägt und zeigen Beispiele für unterschiedliche perfekt leitende Einschlüsse mit gleichen Rückstreudaten.

Influence of fluid and volume state on PaO(2) oscillations in mechanically ventilated pigs

ABSTRACT Varying pulmonary shunt fractions during the respiratory cycle cause oxygen oscillations during mechanical ventilation. In artificially damaged lungs, cyclical recruitment of atelectasis is responsible for varying shunt according to published evidence. We introduce a complimentary hypothesis that cyclically varying shunt in healthy lungs is caused by cyclical redistribution of pulmonary perfusion. Administration of crystalloid or colloid infusions would decrease oxygen oscillations if our hypothesis was right. Therefore, n = 14 mechanically ventilated healthy pigs were investigated in 2 groups: crystalloid (fluid) versus no-fluid administration. Additional volume interventions (colloid infusion, blood withdrawal) were carried out in each pig. Intra-aortal PaO(2) oscillations were recorded using fluorescence quenching technique. Phase shift of oxygen oscillations during altered inspiratory to expiratory (I:E) ventilation ratio and electrical impedance tomography (EIT) served as control methods to exclude that recruitment of atelectasis is responsible for oxygen oscillations. In hypovolemia relevant oxygen oscillations could be recorded. Fluid and volume state changed PaO(2) oscillations according to our hypothesis. Fluid administration led to a mean decline of 105.3 mmHg of the PaO(2) oscillations amplitude (P < 0.001). The difference of the amplitudes between colloid administration and blood withdrawal was 62.4 mmHg in pigs not having received fluids (P = 0.0059). Fluid and volume state also changed the oscillation phase during altered I:E ratio. EIT excluded changes of regional ventilation (i.e., recruitment of atelectasis) to be responsible for these oscillations. In healthy pigs, cyclical redistribution of pulmonary perfusion can explain the size of respiratory-dependent PaO(2) oscillations.

Influence of end-expiratory level and tidal volume on gravitational ventilation distribution during tidal breathing in healthy adults

Our understanding of regional filling of the lung and regional ventilation distribution is based on studies using stepwise inhalation of radiolabelled tracer gases, magnetic resonance imaging and positron emission tomography. We aimed to investigate whether these differences in ventilation distribution at different end-expiratory levels (EELs) and tidal volumes (V (T)s) held also true during tidal breathing. Electrical impedance tomography (EIT) measurements were performed in ten healthy adults in the right lateral position. Five different EELs with four different V (T)s at each EEL were tested in random order, resulting in 19 combinations. There were no measurements for the combination of the highest EEL/highest V (T). EEL and V (T) were controlled by visual feedback based on airflow. The fraction of ventilation directed to different slices of the lung (VENT(RL1)-VENT(RL8)) and the rate of the regional filling of each slice versus the total lung were analysed. With increasing EEL but normal tidal volume, ventilation was preferentially distributed to the dependent lung and the filling of the right and left lung was more homogeneous. With increasing V (T) and maintained normal EEL (FRC), ventilation was preferentially distributed to the dependent lung and regional filling became more inhomogeneous (p < 0.05). We could demonstrate that regional and temporal ventilation distribution during tidal breathing was highly influenced by EEL and V (T).

Regional and overall ventilation inhomogeneities in preterm and term-born infants

OBJECTIVES: We compared ventilation inhomogeneity assessed by electrical impedance tomography (EIT) and multiple breath washout (MBW) in preterm and term-born infants. We hypothesised that EIT measurements in spontaneously breathing infants are repeatable and that differences in regional ventilation distribution measured by EIT can distinguish between preterm and term-born infants. DESIGN: Cross-sectional group comparison study. SETTING: Lung function laboratory at a University Children's Hospital. PARTICIPANTS: Seventeen healthy term-born and 15 preterm infants at a matched postmenstrual age of 44 weeks. MEASUREMENTS AND RESULTS: We concurrently measured ventilation inhomogeneity by EIT, ventilation inhomogeneity (LCI) and functional residual capacity (FRC) by MBW and tidal breathing variables during unsedated quiet sleep. EIT measurements were highly repeatable (coefficient of variation 3.6%). Preterm infants showed significantly more ventilation of the independent parts of the lungs compared to healthy term-born infants assessed by EIT (mean difference 5.0, 95 CI 1.3-8%). Whereas the two groups showed no differences in lung volumes or ventilation inhomogeneities assessed by MBW, EIT discriminated better between term and preterm infants. (FRC/kg: mean difference 1.1 mL, 95% CI -1.4-3.8 mL; LCI: mean difference 0.03, 95% CI -0.32-0.25). CONCLUSIONS: EIT shows distinct differences in ventilation distribution between preterm and term-born infants, which cannot be detected by MBW. Although preterm infants are capable of dynamically maintaining overall functional residual volume and ventilation distribution, they show some spatial differences from fullterm infants.

Effect of different breathing AIDS on ventilation distribution in adults with cystic fibrosis.

BACKGROUND AND OBJECTIVES We investigated the effect of different breathing aids on ventilation distribution in healthy adults and subjects with cystic fibrosis (CF). METHODS In 11 healthy adults and 9 adults with CF electrical impedance tomography measurements were performed during spontaneous breathing, continuous positive airway pressure (CPAP) and positive expiratory pressure (PEP) therapy randomly applied in upright and lateral position. Spatial and temporal ventilation distribution was assessed. RESULTS The proportion of ventilation directed to the dependent lung significantly increased in lateral position compared to upright in healthy and CF. This effect was enhanced with CPAP but neutralised with PEP, whereas the effect of PEP was larger in the healthy group. Temporal ventilation distribution showed exactly the opposite with homogenisation during CPAP and increased inhomogeneity with PEP. CONCLUSIONS PEP shows distinct differences to CPAP with respect to its impact on ventilation distribution in healthy adults and CF subjects EIT might be used to individualise respiratory physiotherapy.

Influence of inspiration to expiration ratio on cyclic recruitment and derecruitment of atelectasis in a saline lavage model of acute respiratory distress syndrome*

OBJECTIVE Cyclic recruitment and derecruitment of atelectasis can occur during mechanical ventilation, especially in injured lungs. Experimentally, cyclic recruitment and derecruitment can be quantified by respiration-dependent changes in PaO2 (ΔPaO2), reflecting the varying intrapulmonary shunt fraction within the respiratory cycle. This study investigated the effect of inspiration to expiration ratio upon ΔPaO2 and Horowitz index. DESIGN Prospective randomized study. SETTING Laboratory investigation. SUBJECTS Piglets, average weight 30 ± 2 kg. INTERVENTIONS At respiratory rate 6 breaths/min, end-inspiratory pressure (Pendinsp) 40 cm H2O, positive end-expiratory pressure 5 cm H2O, and FIO2 1.0, measurements were performed at randomly set inspiration to expiration ratios during baseline healthy and mild surfactant depletion injury. Lung damage was titrated by repetitive surfactant washout to induce maximal cyclic recruitment and derecruitment as measured by multifrequency phase fluorimetry. Regional ventilation distribution was evaluated by electrical impedance tomography. Step changes in airway pressure from 5 to 40 cm H2O and vice versa were performed after lavage to calculate PO2-based recruitment and derecruitment time constants (TAU). MEASUREMENTS AND MAIN RESULTS In baseline healthy, cyclic recruitment and derecruitment could not be provoked, whereas in model acute respiratory distress syndrome, the highest ΔPaO2 were routinely detected at an inspiration to expiration ratio of 1:4 (range, 52-277 torr [6.9-36.9 kPa]). Shorter expiration time reduced cyclic recruitment and derecruitment significantly (158 ± 85 torr [21.1 ± 11.3 kPa] [inspiration to expiration ratio, 1:4]; 25 ± 12 torr [3.3 ± 1.6 kPa] [inspiration to expiration ratio, 4:1]; p < 0.0001), whereas the PaO2/FIO2 ratio increased (267 ± 50 [inspiration to expiration ratio, 1:4]; 424 ± 53 [inspiration to expiration ratio, 4:1]; p < 0.0001). Correspondingly, regional ventilation redistributed toward dependent lung regions (p < 0.0001). Recruitment was much faster (TAU: fast 1.6 s [78%]; slow 9.2 s) than derecruitment (TAU: fast 3.1 s [87%]; slow 17.7 s) (p = 0.0078). CONCLUSIONS Inverse ratio ventilation minimizes cyclic recruitment and derecruitment of atelectasis in an experimental model of surfactant-depleted pigs. Time constants for recruitment and derecruitment, and regional ventilation distribution, reflect these findings and highlight the time dependency of cyclic recruitment and derecruitment.

Correlation between alveolar ventilation and electrical properties of lung parenchyma.

One key problem in modern medical imaging is linking measured data and actual physiological quantities. In this article we derive such a link between the electrical bioimpedance of lung parenchyma, which can be measured by electrical impedance tomography (EIT), and the magnitude of regional ventilation, a key to understanding lung mechanics and developing novel protective ventilation strategies. Two rat-derived three-dimensional alveolar microstructures obtained from synchrotron-based x-ray tomography are each exposed to a constant potential difference for different states of ventilation in a finite element simulation. While the alveolar wall volume remains constant during stretch, the enclosed air volume varies, similar to the lung volume during ventilation. The enclosed air, serving as insulator in the alveolar ensemble, determines the resulting current and accordingly local tissue bioimpedance. From this we can derive a relationship between lung tissue bioimpedance and regional alveolar ventilation. The derived relationship shows a linear dependence between air content and tissue impedance and matches clinical data determined from a ventilated patient at the bedside.

Effect of smoke inhalation on viscoelastic properties and ventilation distribution in sheep

Smoke inhalation injuries are the leading cause of mortality from burn injury. Airway obstruction due to mucus plugging and bronchoconstriction can cause severe ventilation inhomogeneity and worsen hypoxia. Studies describing changes of viscoelastic characteristics of the lung after smoke inhalation are missing. We present results of a new smoke inhalation device in sheep and describe pathophysiological changes after smoke exposure. Fifteen female Merino ewes were anesthetized and intubated. Baseline data using electrical impedance tomography and multiple-breath inert-gas washout were obtained by measuring ventilation distribution, functional residual capacity, lung clearance index, dynamic compliance, and stress index. Ten sheep were exposed to standardized cotton smoke insufflations and five sheep to sham smoke insufflations. Measured carboxyhemoglobin before inhalation was 3.87 +/- 0.28% and 5 min after smoke was 61.5 +/- 2.1%, range 50-69.4% ( P < 0.001). Two hours after smoke functional residual capacity decreased from 1,773 +/- 226 to 1,006 +/- 129 ml and lung clearance index increased from 10.4 +/- 0.4 to 14.2 +/- 0.9. Dynamic compliance decreased from 56.6 +/- 5.5 to 32.8 +/- 3.2 ml/ cmH(2)O. Stress index increased from 0.994 +/- 0.009 to 1.081 +/- 0.011 ( P < 0.01) ( all means +/- SE, P < 0.05). Electrical impedance tomography showed a shift of ventilation from the dependent to the independent lung after smoke exposure. No significant change was seen in the sham group. Smoke inhalation caused immediate onset in pulmonary dysfunction and significant ventilation inhomogeneity. The smoke inhalation device as presented may be useful for interventional studies.

Estudo teórico da tomografia por impedância elétrica

O presente trabalho descreve um estudo sobre a metodologia matemática para a solução do problema direto e inverso na Tomografia por Impedância Elétrica. Este estudo foi motivado pela necessidade de compreender o problema inverso e sua utilidade na formação de imagens por Tomografia por Impedância Elétrica. O entendimento deste estudo possibilitou constatar, através de equações e programas, a identificação das estruturas internas que constituem um corpo. Para isto, primeiramente, é preciso conhecer os potencias elétricos adquiridos nas fronteiras do corpo. Estes potenciais são adquiridos pela aplicação de uma corrente elétrica e resolvidos matematicamente pelo problema direto através da equação de Laplace. O Método dos Elementos Finitos em conjunção com as equações oriundas do eletromagnetismo é utilizado para resolver o problema direto. O software EIDORS, contudo, através dos conceitos de problema direto e inverso, reconstrói imagens de Tomografia por Impedância Elétrica que possibilitam visualizar e comparar diferentes métodos de resolução do problema inverso para reconstrução de estruturas internas. Os métodos de Tikhonov, Noser, Laplace, Hiperparamétrico e Variação Total foram utilizados para obter uma solução aproximada (regularizada) para o problema de identificação. Na Tomografia por Impedância Elétrica, com as condições de contorno preestabelecidas de corrente elétricas e regiões definidas, o método hiperparamétrico apresentou uma solução aproximada mais adequada para reconstrução da imagem.

Data driven regularization models of non-linear ill-posed inverse problems in imaging

Imaging technologies are widely used in application fields such as natural sciences, engineering, medicine, and life sciences. A broad class of imaging problems reduces to solve ill-posed inverse problems (IPs). Traditional strategies to solve these ill-posed IPs rely on variational regularization methods, which are based on minimization of suitable energies, and make use of knowledge about the image formation model (forward operator) and prior knowledge on the solution, but lack in incorporating knowledge directly from data. On the other hand, the more recent learned approaches can easily learn the intricate statistics of images depending on a large set of data, but do not have a systematic method for incorporating prior knowledge about the image formation model. The main purpose of this thesis is to discuss data-driven image reconstruction methods which combine the benefits of these two different reconstruction strategies for the solution of highly nonlinear ill-posed inverse problems. Mathematical formulation and numerical approaches for image IPs, including linear as well as strongly nonlinear problems are described. More specifically we address the Electrical impedance Tomography (EIT) reconstruction problem by unrolling the regularized Gauss-Newton method and integrating the regularization learned by a data-adaptive neural network. Furthermore we investigate the solution of non-linear ill-posed IPs introducing a deep-PnP framework that integrates the graph convolutional denoiser into the proximal Gauss-Newton method with a practical application to the EIT, a recently introduced promising imaging technique. Efficient algorithms are then applied to the solution of the limited electrods problem in EIT, combining compressive sensing techniques and deep learning strategies. Finally, a transformer-based neural network architecture is adapted to restore the noisy solution of the Computed Tomography problem recovered using the filtered back-projection method.