Iterative regularization methods for ill-posed problems

This Ph.D thesis focuses on iterative regularization methods for regularizing linear and nonlinear ill-posed problems. Regarding linear problems, three new stopping rules for the Conjugate Gradient method applied to the normal equations are proposed and tested in many numerical simulations, including some tomographic images reconstruction problems. Regarding nonlinear problems, convergence and convergence rate results are provided for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting.

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.

Regularization meets GreenAI: a new framework for image reconstruction in life sciences applications

Ill-conditioned inverse problems frequently arise in life sciences, particularly in the context of image deblurring and medical image reconstruction. These problems have been addressed through iterative variational algorithms, which regularize the reconstruction by adding prior knowledge about the problem's solution. Despite the theoretical reliability of these methods, their practical utility is constrained by the time required to converge. Recently, the advent of neural networks allowed the development of reconstruction algorithms that can compute highly accurate solutions with minimal time demands. Regrettably, it is well-known that neural networks are sensitive to unexpected noise, and the quality of their reconstructions quickly deteriorates when the input is slightly perturbed. Modern efforts to address this challenge have led to the creation of massive neural network architectures, but this approach is unsustainable from both ecological and economic standpoints. The recently introduced GreenAI paradigm argues that developing sustainable neural network models is essential for practical applications. In this thesis, we aim to bridge the gap between theory and practice by introducing a novel framework that combines the reliability of model-based iterative algorithms with the speed and accuracy of end-to-end neural networks. Additionally, we demonstrate that our framework yields results comparable to state-of-the-art methods while using relatively small, sustainable models. In the first part of this thesis, we discuss the proposed framework from a theoretical perspective. We provide an extension of classical regularization theory, applicable in scenarios where neural networks are employed to solve inverse problems, and we show there exists a trade-off between accuracy and stability. Furthermore, we demonstrate the effectiveness of our methods in common life science-related scenarios. In the second part of the thesis, we initiate an exploration extending the proposed method into the probabilistic domain. We analyze some properties of deep generative models, revealing their potential applicability in addressing ill-posed inverse problems.

Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

Bioimaging of animal embryogenesis: mathematical methods and computational algorithms

Some fundamental biological processes such as embryonic development have been preserved during evolution and are common to species belonging to different phylogenetic positions, but are nowadays largely unknown. The understanding of cell morphodynamics leading to the formation of organized spatial distribution of cells such as tissues and organs can be achieved through the reconstruction of cells shape and position during the development of a live animal embryo. We design in this work a chain of image processing methods to automatically segment and track cells nuclei and membranes during the development of a zebrafish embryo, which has been largely validates as model organism to understand vertebrate development, gene function and healingrepair mechanisms in vertebrates. The embryo is previously labeled through the ubiquitous expression of fluorescent proteins addressed to cells nuclei and membranes, and temporal sequences of volumetric images are acquired with laser scanning microscopy. Cells position is detected by processing nuclei images either through the generalized form of the Hough transform or identifying nuclei position with local maxima after a smoothing preprocessing step. Membranes and nuclei shapes are reconstructed by using PDEs based variational techniques such as the Subjective Surfaces and the Chan Vese method. Cells tracking is performed by combining informations previously detected on cells shape and position with biological regularization constraints. Our results are manually validated and reconstruct the formation of zebrafish brain at 7-8 somite stage with all the cells tracked starting from late sphere stage with less than 2% error for at least 6 hours. Our reconstruction opens the way to a systematic investigation of cellular behaviors, of clonal origin and clonal complexity of brain organs, as well as the contribution of cell proliferation modes and cell movements to the formation of local patterns and morphogenetic fields.

Food lipids: their effects on quality and oxidative stability of animal tissues and emulsions

Lipolysis and oxidation of lipids in foods are the major biochemical and chemical processes that cause food quality deterioration, leading to the characteristic, unpalatable odour and flavour called rancidity. In addition to unpalatability, rancidity may give rise to toxic levels of certain compounds like aldehydes, hydroperoxides, epoxides and cholesterol oxidation products. In this PhD study chromatographic and spectroscopic techniques were employed to determine the degree of lipid oxidation in different animal products and its relationship with technological parameters like feeding fat sources, packaging, processing and storage conditions. To achieve this goal capillary gas chromatography (CGC) was employed not only to determine the fatty acids profile but also, after solid phase extraction, the amount of sterols (cholesterol and phytosterols) and cholesterol oxidation products (COPs). To determine hydroperoxides, primary products of oxidation and quantify secondary products UV/VIS absorbance spectroscopy was applied. Beef and pork meat in this study were analysed. In actual fact, lipid oxidation is a major deterioration reaction in meat, meat products and results in adverse changes in the colour, flavour, texture of meat and develops different compounds which should be a risk to human health as oxysterols. On beef and pork meat, a study of lipid fraction during storage was carried out to evaluate its shelf-life and some nutritional features life saturated/unsaturated fatty acids ratio and sterols content, in according to the interest that has been growing around functional food in the last years. The last part of this research was focused on the study of lipid oxidation in emulsions. In oil-in-water emulsions antioxidant activity of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) was evaluated. The rates of lipid oxidation of 1.0% stripped soybean oil-in-water emulsions with DOPC were followed by monitoring lipid hydroperoxide and hexanal as indicators of primary and secondary oxidation products and the droplet surface charge or zeta potential (ζ) of the emulsions with varying concentrations of DOPC were tested. This manuscript reports the main results obtained in the three activities briefly summarized as follows: 1. study on effects of feeding composition on the photoxidative stability of lipids from beef meat, evaluated during storage under commercial retail conditions; 2. evaluation of effects of diets and storage conditions on the oxidative stability of pork meat lipids; 3. study on oxidative behavior of DOPC in stripped soybean oil-in-water emulsions stabilized by nonionic surfactant.

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.

Multifunctional nanocarriers encapsulating anti-Alzheimer drug for nasal delivery to central nervous system

Alzheimer's disease (AD) is a fatal neurodegenerative condition characterized clinically by progressive memory loss and irreversible cognitive deterioration. It has been shown that there is a progressive degeneration of the brain cholinergic neurons which leads to the appearance of cognitive symptoms of the disease. The aim of this work was the formulation of multifunctional nanocarriers for nasal administration of tacrine-HCl (THA). This route has many advantages; in particular is possible to convey the drug directly to the Central Nervous System, through the olfactory bulb. In particular, were prepared Albumin nanoparticles carrying beta cyclodextrin and two different beta cyclodextrin derivatives (hydroxypropyl beta cyclodextrin and sulphobutylether beta cyclodextrin), and Multifunctional liposomes, prepared using traditional excipients (cholesterol and phosphatidylcholine), partly enriched with α-tocopherol (Toc) and/or polyunsaturated fatty acids (eicosapentaenoic acid and docosahexaenoic acid) (Ω3). Both nanosystems were characterized in terms of size, Zeta potential and encapsulation efficiency. Were also evaluated their functional properties such as mucoadhesion and permeability, using an ex-vivo assay based on nasal sheep mucosa. On Liposomes were also assessed drug neuronal uptake, cell toxicity, antioxidant and, cytoprotective activity in the human neuronal cell line SH-SY5Y and finally tocopherol trans-membrane diffusion. Both the nanocarriers produced presented excellent properties and a high potential as new systems for CNS-delivery of anti-Alzheimer drugs via the nasal route.

Automated Segmentation and Non-Rigid Registration for the Quantitative Evaluation of Myocardial Perfusion from Magnetic Resonance Images

Myocardial perfusion quantification by means of Contrast-Enhanced Cardiac Magnetic Resonance images relies on time consuming frame-by-frame manual tracing of regions of interest. In this Thesis, a novel automated technique for myocardial segmentation and non-rigid registration as a basis for perfusion quantification is presented. The proposed technique is based on three steps: reference frame selection, myocardial segmentation and non-rigid registration. In the first step, the reference frame in which both endo- and epicardial segmentation will be performed is chosen. Endocardial segmentation is achieved by means of a statistical region-based level-set technique followed by a curvature-based regularization motion. Epicardial segmentation is achieved by means of an edge-based level-set technique followed again by a regularization motion. To take into account the changes in position, size and shape of myocardium throughout the sequence due to out of plane respiratory motion, a non-rigid registration algorithm is required. The proposed non-rigid registration scheme consists in a novel multiscale extension of the normalized cross-correlation algorithm in combination with level-set methods. The myocardium is then divided into standard segments. Contrast enhancement curves are computed measuring the mean pixel intensity of each segment over time, and perfusion indices are extracted from each curve. The overall approach has been tested on synthetic and real datasets. For validation purposes, the sequences have been manually traced by an experienced interpreter, and contrast enhancement curves as well as perfusion indices have been computed. Comparisons between automatically extracted and manually obtained contours and enhancement curves showed high inter-technique agreement. Comparisons of perfusion indices computed using both approaches against quantitative coronary angiography and visual interpretation demonstrated that the two technique have similar diagnostic accuracy. In conclusion, the proposed technique allows fast, automated and accurate measurement of intra-myocardial contrast dynamics, and may thus address the strong clinical need for quantitative evaluation of myocardial perfusion.

Local Earthquake Tomography by trans-dimensional Monte Carlo Sampling

In this study a new, fully non-linear, approach to Local Earthquake Tomography is presented. Local Earthquakes Tomography (LET) is a non-linear inversion problem that allows the joint determination of earthquakes parameters and velocity structure from arrival times of waves generated by local sources. Since the early developments of seismic tomography several inversion methods have been developed to solve this problem in a linearized way. In the framework of Monte Carlo sampling, we developed a new code based on the Reversible Jump Markov Chain Monte Carlo sampling method (Rj-McMc). It is a trans-dimensional approach in which the number of unknowns, and thus the model parameterization, is treated as one of the unknowns. I show that our new code allows overcoming major limitations of linearized tomography, opening a new perspective in seismic imaging. Synthetic tests demonstrate that our algorithm is able to produce a robust and reliable tomography without the need to make subjective a-priori assumptions about starting models and parameterization. Moreover it provides a more accurate estimate of uncertainties about the model parameters. Therefore, it is very suitable for investigating the velocity structure in regions that lack of accurate a-priori information. Synthetic tests also reveal that the lack of any regularization constraints allows extracting more information from the observed data and that the velocity structure can be detected also in regions where the density of rays is low and standard linearized codes fails. I also present high-resolution Vp and Vp/Vs models in two widespread investigated regions: the Parkfield segment of the San Andreas Fault (California, USA) and the area around the Alto Tiberina fault (Umbria-Marche, Italy). In both the cases, the models obtained with our code show a substantial improvement in the data fit, if compared with the models obtained from the same data set with the linearized inversion codes.

Strategie formulative per la veicolazione nasale di farmaci

Microparticelle a base di complessi polielettrolitici di Chitosano/Pectina per il rilascio nasale di Tacrina cloridrato. Lo scopo di questo studio è stata la ricerca di nuove formulazioni solide per la somministrazione nasale di Tacrina cloridrato allo scopo di ridurre l’eccessivo effetto di primo passaggio epatico ed aumentarne la biodisponibilità a livello del Sistema Nervoso Centrale. La Tacrina è stata incapsulata in microparticelle mucoadesive a base di complessi elettrolitici di chitosano e pectina. Le microparticelle sono state preparate mediante due diversi approcci tecnologici (spray-drying e spray-drying/liofilizzazione) e analizzate in termini di caratteristiche dimensionali, morfologiche e chimico-fisiche. Nanoparticelle di Chitosano reticolate con Sodio Cromoglicato per il trattamento della rinite allergica. Il Sodio Cromoglicato è uno dei farmaci utilizzati per il trattamento della rinite allergica. Come noto, la clearance mucociliare provoca una rapida rimozione dei farmaci in soluzione dalla cavità nasale, aumentando così il numero di somministrazioni giornaliere e, di conseguenza, riducendo la compliance del paziente. Per ovviare a tale problema, si è pensato di includere il sodio cromoglicato in nanoparticelle di chitosano, un polimero capace di aderire alla mucosa nasale, prolungare il contatto della formulazione con il sito di applicazione e ridurre il numero di somministrazioni giornaliere. Le nanoparticelle ottenute sono state caratterizzate in termini di dimensioni, resa, efficienza di incapsulazione e caricamento del farmaco, potenziale zeta e caratteristiche mucoadesive. Analisi quantitativa di Budesonide amorfa tramite calorimetria a scansione differenziale. È stato sviluppato un nuovo metodo quantitativo allo stato solido basato sulla Calorimetria a Scansione Differenziale (DSC) in grado di quantificare in modo selettivo e accurato la quantità di Budesonide amorfa presente in una miscela solida. Durante lo sviluppo del metodo sono stati affrontati problemi relativi alla convalida di metodi analitici su campioni solidi quali la miscelazione di polveri solide per la preparazione di miscele standard e il calcolo della precisione.