<div>There is an increasing need to accurately image objects at a high temporal resolution for different applications in order to analyze the underlying physical, chemical, or biological processes. In this thesis, we use advanced models exploiting the image structure and the measurement process in order to achieve an improved temporal resolution. The thesis is divided into three chapters, each corresponding to a different imaging application.</div><div><br></div><div>In the first chapter, we propose a novel method to localize neurons in fluorescence microscopy images. Accurate localization of neurons enables us to scan only the neuron locations instead of the full brain volume and thus improve the temporal resolution of neuron activity monitoring. We formulate the neuron localization problem as an inverse problem where we reconstruct an image that encodes the location of the neuron centers. The sparsity of the neuron centers serves as a prior model, while the forward model comprises of shape models estimated from training data.</div><div><br></div><div>In the second chapter, we introduce multi-slice fusion, a novel framework to incorporate advanced prior models for inverse problems spanning many dimensions such as 4D computed tomography (CT) reconstruction. State of the art 4D reconstruction methods use model based iterative reconstruction (MBIR), but it depends critically on the quality of the prior modeling. Incorporating deep convolutional neural networks (CNNs) in the 4D reconstruction problem is difficult due to computational difficulties and lack of high-dimensional training data. Multi-Slice Fusion integrates the tomographic forward model with multiple low dimensional CNN denoisers along different planes to produce a 4D regularized reconstruction. The improved regularization in multi-slice fusion allows each time-frame to be reconstructed from fewer measurements, resulting in an improved temporal resolution in the reconstruction. Experimental results on sparse-view and limited-angle CT data demonstrate that Multi-Slice Fusion can substantially improve the quality of reconstructions relative to traditional methods, while also being practical to implement and train.</div><div><br></div><div>In the final chapter, we introduce CodEx, a synergistic combination of coded acquisition and a non-convex Bayesian reconstruction for improving acquisition speed in computed tomography (CT). In an ideal ``step-and-shoot'' tomographic acquisition, the object is rotated to each desired angle, and the view is taken. However, step-and-shoot acquisition is slow and can waste photons, so in practice the object typically rotates continuously in time, leading to views that are blurry. This blur can then result in reconstructions with severe motion artifacts. CodEx works by encoding the acquisition with a known binary code that the reconstruction algorithm then inverts. The CodEx reconstruction method uses the alternating direction method of multipliers (ADMM) to split the inverse problem into iterative deblurring and reconstruction sub-problems, making reconstruction practical. CodEx allows for a fast data acquisition leading to a good temporal resolution in the reconstruction.</div>

Non-destructive imaging modalities for evaluating the internal properties of materials can be formulated as physics-driven inverse problems. Model-based Iterative reconstruction (MBIR) methods that integrate a forward model of the imaging system and a prior model of the object being imaged can provide superior reconstruction quality relative to conventional methods. However, making MBIR feasible for practical applications faces two key challenges. First, we require efficient computational methods for MBIR that allow large-scale reconstructions in real-time. Second, we must develop forward models that accurately capture the physics and geometry of the imaging system, and, support the use of advanced denoisers that enhance image quality as prior models.<br><br>This thesis attempts to address the aforementioned challenges and is divided into three main chapters, each corresponding to a different inverse imaging application. <br><br>In the first chapter of this thesis, we propose a novel 4D model-based iterative reconstruction (MBIR) algorithm for low-angle coherent-scatter X-ray Diffraction (XRD) tomography that can substantially increase the SNR. Our forward model is based on a Poisson photon counting model that incorporates a spatial point-spread function, detector energy response and energy-dependent attenuation correction. Our prior model uses a Markov random field (MRF) together with a reduced spectral bases set determined using non-negative matrix factorization. Our algorithm efficiently computes the Bayesian estimate by exploiting the sparsity of the measurement data. We demonstrate the ability of our method to achieve sufficient spatial resolution from sparse photon-starved measurements and also discriminate between materials of similar densities with real datasets.<br><br>In the second chapter of this thesis, we propose a multi-agent consensus equilibrium (MACE) algorithm for distributing both the computation and memory of <br>MBIR for Computed Tomographic (CT) reconstruction across a large number of parallel nodes. In MACE, each node stores only a sparse subset of views and a small portion of the system matrix, and each parallel node performs a local sparse-view reconstruction, which based on repeated feedback from other nodes, converges to the global optimum. Our distributed approach can also incorporate advanced denoisers as priors to enhance reconstruction quality. In this case, we obtain a parallel solution to the serial framework of Plug-n-play (PnP) priors, which we call MACE-PnP. In order to make MACE practical, we introduce a partial update method that eliminates nested iterations and prove that it converges to the same global solution. Finally, we validate our approach on a distributed memory system with real CT data. We also demonstrate an implementation of our approach on a massive supercomputer that can perform large-scale reconstruction in real-time. <br><br>In the third chapter of this thesis, we propose a method that makes MBIR feasible for real-time single-shot holographic imaging through deep turbulence. Our method uses surrogate optimization techniques to simplify and speedup the reflectance and phase-error updates in MBIR. Further, our method accelerates computation of the surrogate-updates by leveraging cache-prefetching and SIMD vector processing units on a single CPU core. We analyze the convergence and real CPU time of our method using simulated datasets, and demonstrate its dramatic speedup over the original MBIR approach. <br>

In this paper we deal with aspects of characterizing the ill-posedn ess of nonlinear inverse problems based on the discussion of specific examples. In particular, a parameter identification problem to a second order differential equation and its ill-posed linear components are under consideration. A new approach to the classification ofill-posedness degrees for multiplication operators completes the paper.

The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a variant of the discrepancy principle is analyzed. In many cases such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

Title: Numerical Methods in Discrete Inverse Problems Author: Marie Kubínová Department: Department of Numerical Mathematics Supervisor: RNDr. Iveta Hnětynková, Ph.D., Department of Numerical Mathe- matics Abstract: Inverse problems represent a broad class of problems of reconstruct- ing unknown quantities from measured data. A common characteristic of these problems is high sensitivity of the solution to perturbations in the data. The aim of numerical methods is to approximate the solution in a computationally efficient way while suppressing the influence of inaccuracies in the data, referred to as noise, that are always present. Properties of noise and its behavior in reg- ularization methods play crucial role in the design and analysis of the methods. The thesis focuses on several aspects of solution of discrete inverse problems, in particular: on propagation of noise in iterative methods and its representation in the corresponding residuals, including the study of influence of finite-precision computation, on estimating the noise level, and on solving problems with data polluted with noise coming from various sources. Keywords: discrete inverse problems, iterative solvers, noise estimation, mixed noise, finite-precision arithmetic - iii -

Les problèmes de complétion de données interviennent dans divers domaines de la physique, tels que la mécanique, l'acoustique ou la thermique. La mesure directe des conditions aux limites se heurte souvent à l'impossibilité de placer l'instrumentation adéquate. La détermination de ces données n'est alors possible que grâce à des informations complémentaires. Des mesures surabondantes sur une partie accessible de la frontière mènent à la résolution d'un problème inverse de type Cauchy. Cependant, dans certains cas, des mesures directes sur la frontière sont irréalisables, des mesures de champs plus facilement accessibles permettent de pallier ce problème. Cette thèse présente des méthodes de régularisation évanescente qui permettent de trouver, parmi toutes les solutions de l'équation d'équilibre, la solution du problème de complétion de données qui s'approche au mieux des données de type Cauchy ou de champs partiels. Ces processus itératifs ne dépendent pas d'un coefficient de régularisation et sont robustes vis à vis du bruit sur les données, qui sont recalculées et de ce fait débruitées. Nous nous intéressons, dans un premier temps, à la résolution de problèmes de Cauchy associés à l'équation d'Helmholtz. Une étude numérique complète est menée, en utilisant la méthode des solutions fondamentales en tant que méthode numérique pour discrétiser l'espace des solutions de l'équation d'Helmholtz. Des reconstructions précises attestent de l'efficacité et de la robustesse de la méthode. Nous présentons, dans un second temps, la généralisation de la méthode de régularisation évanescente aux problèmes de complétion de données à partir de mesures de champs partielles. Des simulations numériques, pour l'opérateur de Lamé, dans le cadre des éléments finis et des solutions fondamentales, montrent la capacité de la méthode à compléter et débruiter des données partielles de champs de déplacements et à identifier les conditions aux limites en tout point de la frontière. Nous retrouvons des reconstructions précises et un débruitage efficace des données lorsque l'algorithme est appliqué à des mesures réelles issues de corrélation d'images numériques. Un éventuel changement de comportement du matériau est détecté grâce à l'analyse des résidus de déplacements. / Data completion problems occur in many engineering fields, such as mechanical, acoustical and thermal sciences. Direct measurement of boundary conditions is often confronting with the impossibility of placing the appropriate instrumentation. The determination of these data is then possible only through additional informations. Overprescribed measurements on an accessible part of the boundary lead to the resolution of an inverse Cauchy problem. However, in some cases, direct measurements on the boundary are inaccessible, to overcome this problem field measurements are more easily accessible. This thesis presents fading regularization methods that allow to find, among all the solutions of the equilibrium equation, the solution of the data completion problem which fits at best Cauchy or partial fields data. These iterative processesdo not depend on a regularization coefficient and are robust with respect to the noise on the data, which are recomputed and therefore denoised. We are interested initially in solving Cauchy problems associated with the Helmholtz equation. A complete numerical study is made, usingthe method of fundamental solutions as a numerical method for discretizing the space of the Helmholtz equation solutions. Accurate reconstructions attest to the efficiency and the robustness of the method. We present, in a second time, the generalization of the fading regularization method to the data completion problems from partial full-field measurements. Numerical simulations, for the Lamé operator, using the finite element method or the method of fundamental solutions, show the ability of the iterative process to complete and denoise partial displacements fields data and to identify the boundary conditions at any point. We find precise reconstructions and efficient denoising of the data when the algorithm is applied to real measurements from digital image correlation. A possible change in the material behavior is detected thanks to the analysis of the displacements residuals.

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Develop new strategies to design and operate the multilateration systems, used for air traffic control operations, in a more efficient way. The design strategies are based on the utilization of metaheuristic optimization techniques and they are intended to found the optimal spatial distribution of the system ground stations, taking into account the most relevant system operation parameters. The strategies to operate the systems are based on the development of new positioning methods which allow solving the problems of uncertainty position and poor accuracy that the current systems can present. The new strategies can be applied to design, deploy and operate the multilateration systems for airport surface surveillance as well as takeoff-landing, approach and enroute control. An important advance in the current knowledge of air traffic control is expected from the development of these strategies, because they solve several deficiencies that have been made clear, by the international scientific community, in the last years. / Mantilla Gaviria, IA. (2013). New Strategies to Improve Multilateration Systems in the Air Traffic Control [Tesis doctoral]. Editorial Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/29688 / Alfresco

[ES] El tema de la investigación se centra en técnicas avanzadas para manejar problemas de aguas subterráneas y superficiales relacionados con métodos inversos y cambio climático. Los filtros de Kalman, con especial atención en Ensemble Smoother with Multiple Data Assimilation (ES-MDA), se analizan y mejoran para la solución de diferentes tipos de problemas inversos. En particular, la principal novedad es la aplicación de estos métodos para la identificación de series temporales. La primera parte de la tesis, luego de la descripción del método, presenta el desarrollo de un software escrito en Python para la aplicación de la metodología propuesta. El software cuenta con un flujo de trabajo flexible que puede adaptarse fácilmente para implementar diferentes variantes del filtro de Kalman y ser aplicado para la solución de varios tipos de problemas. Un paquete de herramientas proporciona varias funcionalidades que permiten de configurar el algoritmo de acuerdo con el problema específico analizado. La primera aplicación se refiere a la solución del problema inverso de flujo en ríos. Este es un procedimiento inverso destinado a estimar el flujo de entrada a un sistema hidráulico en función de información recopilada abajo. El procedimiento se prueba mediante dos ejemplos sintéticos y un estudio de caso real; se investiga el impacto de los tamaños de los conjuntos y la aplicación de técnicas de localización e inflación de covarianzas. Los resultados muestran la capacidad del método propuesto de resolver este tipo de problemas; el rendimiento de ES-MDA mejora, especialmente para tamaños de conjuntos pequeños, cuando se aplican técnicas de inflación y localización de covarianza. La segunda aplicación en el campo de las aguas superficiales se refiere a la calibración de un modelo hidrológico-hidráulico que simula los mecanismos de formación de eventos de inundación. ES-MDA se acopla al modelo numérico de forma paralela para la estimación de los coeficientes de rugosidad e infiltración en base al conocimiento de un hidrograma de flujo en una sección del dominio. Los resultados de dos casos sintéticos y un estudio de caso real demuestran la capacidad del método propuesto para calibrar el modelo hidrológico-hidráulico con un tiempo computacional razonable. En el campo de aguas subterráneas, ES-MDA se aplica por primera vez para identificar simultáneamente la ubicación de la fuente y el historial de liberación de un contaminante en un acuífero a partir de datos de concentración detectados en diferentes puntos del dominio. Se realizaron numerosas pruebas para evaluar la influencia de la distribución espacial y temporal de los datos de concentración, el número del conjunto y el uso de técnicas de localización e inflación; además, se presenta un nuevo procedimiento para realizar una localización iterativa espacio-temporal. La metodología se valida mediante un ejemplo analítico y un estudio de caso que utiliza datos obtenidos en el laboratorio mediante una caja de arena. ES-MDA conduce a una buena estimación de los parámetros investigados; una red de monitoreo bien diseñada y la aplicación de correcciones de covarianza mejoran el rendimiento del método y ayudan a mitigar el posible problema de no unicidad de la solución. Otro propósito de la tesis es investigar el efecto del cambio climático en las aguas subterráneas. Se presenta un modelo simplificado que describe la respuesta de los niveles de agua subterránea a las variables meteorológicas hasta 2100. Es un enfoque estadístico sencillo basado en las correlaciones entre los niveles de agua subterránea y dos índices de sequía que dependen de los datos de precipitación y temperatura. El método se utiliza para evaluar el impacto del cambio climático en los recursos de agua subterránea en un área de estudio ubicada en el norte de Italia utilizando datos históricos y de modelos climáticos regionales. Los resultados m / [CA] El tema de la investigació se centra en tècniques avançades per a manejar problemes d'aigües subterrànies i superficials relacionats amb mètodes inversos i canvi climàtic. Els filtres de Kalman, amb especial atenció en Ensemble Smoother with Multiple Data Assimilation (ES-MDA), s'analitzen i milloren per a la solució de diferents tipus de problemes inversos. En particular, la principal novetat és l'aplicació d'aquests mètodes per a la identificació de sèries temporals. La primera part de la tesi presenta el desenvolupament d'un programari escrit en Python per l'aplicació de la metodologia presentada. El programari compta amb un flux de treball flexible que pot adaptar-se fàcilment per a implementar diferents variants del filtre de Kalman i ser aplicat per a la solució de diversos tipus de problemes. Un paquet complementar d'eines proporciona diverses funcionalitats que permeten de configurar l'algorisme d'acord amb el problema específic analitzat. La primera aplicació es un nou enfocament per la solució del problema invers de flux en rius. Aquest és un procediment invers destinat a estimar el flux d'entrada a un sistema hidràulic en funció d'informació recopilada aigües avall. El procediment es prova mitjançant dos exemples sintètics i un estudi de cas real; s'investiga l'impacte de les grandàries dels conjunts i l'aplicació de tècniques de localització i inflació de covariàncies. Els resultats mostren la capacitat del mètode proposat de resoldre aquest tipus de problemes; el rendiment de ES-MDA millora, especialment per a grandàries de conjunts xicotets, quan s'apliquen tècniques d'inflació i localització de covariància. La segona aplicació en el camp de les aigües superficials es refereix al calibratge d'un model hidrològic-hidràulic que simula els mecanismes de formació d'esdeveniments d'inundació a partir de sollicitació hidrometeorológicas i la seua posterior propagació. ES-MDA s'acobla al model numèric de manera paral·lela per l'estimació dels coeficients de rugositat i infiltració sobre la base del coneixement d'un hidrograma de flux en una secció del domini. Els resultats de dos casos sintètics i un estudi de cas real demostren la capacitat del mètode proposat per calibrar el model hidrològic-hidràulic amb un temps computacional raonable. En el camp d'aigües subterrànies, ES-MDA s'aplica per primera vegada per identificar simultàniament la ubicació de la font i l'historial d'alliberament d'un contaminant en un aqüífer a partir d'un conjunt de dades de concentració detectats en diferents punts del domini. Es van realitzar nombroses proves per avaluar la influència de la distribució espacial i temporal de les dades de concentració, el número del conjunt i l'ús de tècniques de localització i inflació; a més, es presenta un nou procediment per realitzar una localització iterativa espaciotemporal. La metodologia es valguda mitjançant un exemple analític i un estudi de cas per al qual s'utilitzen dades obtingudes en el laboratori mitjançant una caixa d'arena. ES-MDA condueix a una bona reconstrucció dels paràmetres investigats; una xarxa de monitoratge ben dissenyada i l'aplicació de correccions de covariància milloren el rendiment del mètode i ajuden a mitigar el possible problema de no unicitat de la solució. Un altre propòsit de la tesi és investigar l'efecte del canvi climàtic en les aigües subterrànies. Es presenta un model simplificat que descriu la resposta dels nivells d'aigua subterrània a les variables meteorològiques fins a 2100. És un enfocament estadístic senzill basat en les correlacions entre els nivells d'aigua subterrània i dos índexs de sequera que depenen de les dades de precipitació i temperatura. El mètode s'utilitza per a avaluar l'impacte del canvi climàtic en els recursos d'aigua subterrània en una àrea d'estudi situada en el nord d'Itàlia utilitzant dades històriques i de models climàtics regionals. / [EN] This work focuses on the investigation of advanced techniques to handle groundwater and surface water problems in the framework of inverse methods and climate change. The Ensemble Kalman filter methods, with particular attention to the Ensemble Smoother with Multiple Data Assimilation (ES-MDA), are extensively analyzed and improved for the solution of different types of inverse problems. In particular, the main novelty is the application of these methods for the identification of time series function. In the first part of the thesis, after the description of the ES-MDA method, the development of a Python software package for the application of the proposed methodology is presented. It is designed with a flexible workflow that can be easily adapted to implement different variants of the Ensemble Kalman filter and to be applied for the solution of various types of inverse problems. A complemented tool package provides several functionalities that allow to setup the algorithm configuration suiting the specific analyzed problem. The first novelty application of the ES-MDA method aimed at solving the reverse flow routing problem. The objective of the inverse procedure is the estimation of an unknown inflow hydrograph to a hydraulic system on the basis of information collected downstream and a given forward routing model that relates inflow hydrograph and downstream observations. The procedure is tested by means of two synthetic examples and a real case study; the impact of ensemble sizes and the application of covariance localization and inflation techniques are also investigated. The tests show the capability of the proposed method to solve this type of problem; the performance of ES-MDA improves, especially for small ensemble sizes, when covariance localization and inflation techniques are applied. The second application, in the context of surface water, concerns the calibration of a hydrological-hydraulic model that simulates rainfall-runoff processes. The ES-MDA is coupled with the numerical model by parallel way for the estimation of roughness and infiltration coefficients based on the knowledge of a discharge hydrograph at the basin outlet. The results of two synthetic tests and a real case study demonstrate the capability of the proposed method to calibrate the hydrological-hydraulic model with a reasonable computational time. In the groundwater field, ES-MDA is applied for the first time to simultaneously identify the source location and the release history of a contaminant spill in an aquifer from a sparse set of concentration data collected in few points of the aquifer. The impacts of the concentration sampling scheme, the ensemble size and the use of covariance localization and covariance inflation techniques are tested; furthermore, a new procedure to perform a spatiotemporal iterative localization is presented. The methodology is tested by means of an analytical example and a study case that uses real data collected in a laboratory sandbox. ES-MDA leads to a good estimation of the investigated parameters; a well-designed monitoring network and the use of covariance corrections improve the performance of the method and help to minimize ill-posedness and equifinality. A part of the thesis investigates the impact of climate change on the groundwater availability. A surrogate model that describes the response of groundwater levels to meteorological variables up to 2100 is presented. It is a simple statistical approach based on the correlations between groundwater levels and two drought indices that depend on precipitation and temperature data. The presented method is used to evaluate the impact of climate change on groundwater resources in a study area located in Northern Italy using historical and regional climate model data. The results denote a progressive increase of groundwater droughts in the investigated area. / Todaro, V. (2021). Advanced techniques for solving groundwater and surface water problems in the context of inverse methods and climate change [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/166439 / TESIS

Reconstruction of sonar images is an inverse problem, which is normally solved with model-based methods. These methods may introduce undesired artifacts called angular and range leakage into the reconstruction. In this thesis, a method called Learned Primal-Dual Reconstruction, which combines a data-driven and a model-based approach, is used to investigate the use of data-driven methods for reconstruction within sonar imaging. The method uses primal and dual variables inspired by classical optimization methods where parts are replaced by convolutional neural networks to iteratively find a solution to the reconstruction problem. The network is trained and validated with synthetic data on eight models with different architectures and training parameters. The models are evaluated on measurement data and the results are compared with those from a purely model-based method. Reconstructions performed on synthetic data, where a ground truth image is available, show that it is possible to achieve reconstructions with the data-driven method that have less leakage than reconstructions from the model-based method. For reconstructions performed on measurement data where no ground truth is available, some variants of the learned model achieve a good result with less leakage.