Cybersecurity and Quantum Computing: friends or foes?

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An evaluation of partial differential equations based digital inpainting algorithms

Partial Differential equations (PDEs) have been used to model various phenomena/tasks in different scientific and engineering endeavours. This thesis is devoted to modelling image inpainting by numerical implementations of certain PDEs. The main objectives of image inpainting include reconstructing damaged parts and filling-in regions in which data/colour information are missing. Different automatic and semi-automatic approaches to image inpainting have been developed including PDE-based, texture synthesis-based, exemplar-based, and hybrid approaches. Various challenges remain unresolved in reconstructing large size missing regions and/or missing areas with highly textured surroundings. Our main aim is to address such challenges by developing new advanced schemes with particular focus on using PDEs of different orders to preserve continuity of textural and geometric information in the surrounding of missing regions. We first investigated the problem of partial colour restoration in an image region whose greyscale channel is intact. A PDE-based solution is known that is modelled as minimising total variation of gradients in the different colour channels. We extend the applicability of this model to partial inpainting in other 3-channels colour spaces (such as RGB where information is missing in any of the two colours), simply by exploiting the known linear/affine relationships between different colouring models in the derivation of a modified PDE solution obtained by using the Euler-Lagrange minimisation of the corresponding gradient Total Variation (TV). We also developed two TV models on the relations between greyscale and colour channels using the Laplacian operator and the directional derivatives of gradients. The corresponding Euler-Lagrange minimisation yields two new PDEs of different orders for partial colourisation. We implemented these solutions in both spatial and frequency domains. We measure the success of these models by evaluating known image quality measures in inpainted regions for sufficiently large datasets and scenarios. The results reveal that our schemes compare well with existing algorithms, but inpainting large regions remains a challenge. Secondly, we investigate the Total Inpainting (TI) problem where all colour channels are missing in an image region. Reviewing and implementing existing PDE-based total inpainting methods reveal that high order PDEs, applied to each colour channel separately, perform well but are influenced by the size of the region and the quantity of texture surrounding it. Here we developed a TI scheme that benefits from our partial inpainting approach and apply two PDE methods to recover the missing regions in the image. First, we extract the (Y, Cb, Cr) of the image outside the missing region, apply the above PDE methods for reconstructing the missing regions in the luminance channel (Y), and then use the colourisation method to recover the missing (Cb, Cr) colours in the region. We shall demonstrate that compared to existing TI algorithms, our proposed method (using 2 PDE methods) performs well when tested on large datasets of natural and face images. Furthermore, this helps understanding of the impact of the texture in the surrounding areas on inpainting and opens new research directions. Thirdly, we investigate existing Exemplar-Based Inpainting (EBI) methods that do not use PDEs but simultaneously propagate the texture and structure into the missing region by finding similar patches within the rest of image and copying them into the boundary of the missing region. The order of patch propagation is determined by a priority function, and the similarity is determined by matching criteria. We shall exploit recently emerging Topological Data Analysis (TDA) tools to create innovative EBI schemes, referred to as TEBI. TDA studies shapes of data/objects to quantify image texture in terms of connectivity and closeness properties of certain data landmarks. Such quantifications help determine the appropriate size of patch propagation and will be used to modify the patch propagation priority function using the geometrical properties of curvature of isophotes, and to improve the matching criteria of patches by calculating the correlation coefficients from the spatial, gradient and Laplacian domains. The performance of this TEBI method will be tested by applying it to natural dataset images, resulting in improved inpainting when compared with other EBI methods. Fourthly, the recent hybrid-based inpainting techniques are reviewed and a number of highly performing innovative hybrid techniques that combine the use of high order PDE methods with the TEBI method for the simultaneous rebuilding of the missing texture and structure regions in an image are proposed. Such a hybrid scheme first decomposes the image into texture and structure components, and then the missing regions in these components are recovered by TEBI and PDE based methods respectively. The performance of our hybrid schemes will be compared with two existing hybrid algorithms. Fifthly, we turn our attention to inpainting large missing regions, and develop an innovative inpainting scheme that uses the concept of seam carving to reduce this problem to that of inpainting a smaller size missing region that can be dealt with efficiently using the inpainting schemes developed above. Seam carving resizes images based on content-awareness of the image for both reduction and expansion without affecting those image regions that have rich information. The missing region of the seam-carved version will be recovered by the TEBI method, original image size is restored by adding the removed seams and the missing parts of the added seams are then repaired using a high order PDE inpainting scheme. The benefits of this approach in dealing with large missing regions are demonstrated. The extensive performance testing of the developed inpainting methods shows that these methods significantly outperform existing inpainting methods for such a challenging task. However, the performance is still not acceptable in recovering large missing regions in high texture and structure images, and hence we shall identify remaining challenges to be investigated in the future. We shall also extend our work by investigating recently developed deep learning based image/video colourisation, with the aim of overcoming its limitations and shortcoming. Finally, we should also describe our on-going research into using TDA to detect recently growing serious “malicious” use of inpainting to create Fake images/videos

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In dieser Arbeit werden spektral kodierte multispektrale Lichtfelder untersucht, wie sie von einer Lichtfeldkamera mit einem spektral kodierten Mikrolinsenarray aufgenommen werden. Für die Rekonstruktion der kodierten Lichtfelder werden zwei Methoden entwickelt, eine basierend auf den Prinzipien des Compressed Sensing sowie eine Deep Learning Methode. Anhand neuartiger synthetischer und realer Datensätze werden die vorgeschlagenen Rekonstruktionsansätze im Detail evaluiert

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In this work, spatio-spectrally coded multispectral light fields, as taken by a light field camera with a spectrally coded microlens array, are investigated. For the reconstruction of the coded light fields, two methods, one based on the principles of compressed sensing and one deep learning approach, are developed. Using novel synthetic as well as a real-world datasets, the proposed reconstruction approaches are evaluated in detail

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In dieser Arbeit werden spektral codierte multispektrale Lichtfelder, wie sie von einer Lichtfeldkamera mit einem spektral codierten Mikrolinsenarray aufgenommen werden, untersucht. Für die Rekonstruktion der codierten Lichtfelder werden zwei Methoden entwickelt und im Detail ausgewertet. Zunächst wird eine vollständige Rekonstruktion des spektralen Lichtfelds entwickelt, die auf den Prinzipien des Compressed Sensing basiert. Um die spektralen Lichtfelder spärlich darzustellen, werden 5D-DCT-Basen sowie ein Ansatz zum Lernen eines Dictionary untersucht. Der konventionelle vektorisierte Dictionary-Lernansatz wird auf eine tensorielle Notation verallgemeinert, um das Lichtfeld-Dictionary tensoriell zu faktorisieren. Aufgrund der reduzierten Anzahl von zu lernenden Parametern ermöglicht dieser Ansatz größere effektive Atomgrößen. Zweitens wird eine auf Deep Learning basierende Rekonstruktion der spektralen Zentralansicht und der zugehörigen Disparitätskarte aus dem codierten Lichtfeld entwickelt. Dabei wird die gewünschte Information direkt aus den codierten Messungen geschätzt. Es werden verschiedene Strategien des entsprechenden Multi-Task-Trainings verglichen. Um die Qualität der Rekonstruktion weiter zu verbessern, wird eine neuartige Methode zur Einbeziehung von Hilfslossfunktionen auf der Grundlage ihrer jeweiligen normalisierten Gradientenähnlichkeit entwickelt und gezeigt, dass sie bisherige adaptive Methoden übertrifft. Um die verschiedenen Rekonstruktionsansätze zu trainieren und zu bewerten, werden zwei Datensätze erstellt. Zunächst wird ein großer synthetischer spektraler Lichtfelddatensatz mit verfügbarer Disparität Ground Truth unter Verwendung eines Raytracers erstellt. Dieser Datensatz, der etwa 100k spektrale Lichtfelder mit dazugehöriger Disparität enthält, wird in einen Trainings-, Validierungs- und Testdatensatz aufgeteilt. Um die Qualität weiter zu bewerten, werden sieben handgefertigte Szenen, so genannte Datensatz-Challenges, erstellt. Schließlich wird ein realer spektraler Lichtfelddatensatz mit einer speziell angefertigten spektralen Lichtfeldreferenzkamera aufgenommen. Die radiometrische und geometrische Kalibrierung der Kamera wird im Detail besprochen. Anhand der neuen Datensätze werden die vorgeschlagenen Rekonstruktionsansätze im Detail bewertet. Es werden verschiedene Codierungsmasken untersucht -- zufällige, reguläre, sowie Ende-zu-Ende optimierte Codierungsmasken, die mit einer neuartigen differenzierbaren fraktalen Generierung erzeugt werden. Darüber hinaus werden weitere Untersuchungen durchgeführt, zum Beispiel bezüglich der Abhängigkeit von Rauschen, der Winkelauflösung oder Tiefe. Insgesamt sind die Ergebnisse überzeugend und zeigen eine hohe Rekonstruktionsqualität. Die Deep-Learning-basierte Rekonstruktion, insbesondere wenn sie mit adaptiven Multitasking- und Hilfslossstrategien trainiert wird, übertrifft die Compressed-Sensing-basierte Rekonstruktion mit anschließender Disparitätsschätzung nach dem Stand der Technik

Reconstruction from Spatio-Spectrally Coded Multispectral Light Fields

In this work, spatio-spectrally coded multispectral light fields, as taken by a light field camera with a spectrally coded microlens array, are investigated. For the reconstruction of the coded light fields, two methods, one based on the principles of compressed sensing and one deep learning approach, are developed. Using novel synthetic as well as a real-world datasets, the proposed reconstruction approaches are evaluated in detail

Real-space simulation of two-dimensional interacting quantum condensed matter systems

The proliferation of quantum fluctuations and long-range entanglement presents an outstanding challenge for the numerical simulation of quantum condensed matter systems with exotic ground states. In this thesis, I tackle two classes of two-dimensional interacting models on the honeycomb lattice: multi-orbital Hubbard models on zig zag transition metal dichalcogenide nanoribbons and generalised Kitaev models on periodic clusters. In the first part of the thesis, I discuss novel results obtained in a comparative study of mean field theory (MFT) and determinant quantum Monte Carlo (DetQMC). MFT reveals the influence of the edge filling on the ground state of the ribbons. The unbiased, numerically exact DetQMC confirms the stability of one of the possible ground states, albeit with quantitative differences, such as the critical Hubbard interaction for the onset of magnetic order. Unfortunately, DetQMC is severely plagued by the sign problem for this model. The variance of its estimators grows exponentially as most of the relevant edge fillings are reached and simulations are rendered unfeasibly expensive from the computational standpoint. Motivated by the difficulties posed by the sign problem, I carry out a survey of general purpose numerical methods. The second part of the thesis addresses quantum spin liquids — which have attracted increasing attention — presenting a toolset of Chebyshev spectral methods developed here, namely: the finite temperature Chebyshev polynomial and the hybrid Lanczos-Chebyshev methods. The first one enables studies of temperature dependence for quantities of experimental interest, such as the specific heat, with a two-fold speed-up with respect to state-of-the-art methods. The second one gives access to spectral functions efficiently and with unparalleled flexibility. I use it to obtain novel results for the spin susceptibility of the Kitaev-Ising model, unravelling dynamical signatures of a liquid–to–liquid transition. Finally, I briefly discuss the integration of the novel Chebyshev toolset with existing open-source software

Razonamiento regresivo en situaciones de resolución de problemas: un modelo multidimensional

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 20-10-20The increasing technological progress has highlighted the importance of problem-solving processes and skills connected to programming methods. Among them, backward reasoning is recognized as a critical issue in advanced mathematics education. This, together with the growing interest in recent years of game-based university education is at the base of this research project. Two objectives are established: on the one hand, to extend the epistemic model of backward reasoning, existing in the mathematical literature, to a cognitive and didactic one; on the other hand, to establish principles for the design of university teaching situations focused on backward reasoning. To reach these objectives, four design experiments using strategy games and mathematical problems are developed. These involved a total of 322 university students, from first year of bachelor to PhD, attending the Universidad Complutense de Madrid (Spain) and the Università di Torino (Italy). They are involved in scientific careers (Mathematics, Mathematics Engineering and Computer Science) and teacher training careers (future mathematics professors in secondary school)...El creciente progreso tecnológico ha puesto de relieve la importancia de los procesos de resolución de problemas y los conocimientos técnicos relacionados con los métodos de programación. Entre ellos, el razonamiento regresivo se reconoce como una cuestión crítica en la enseñanza de las matemáticas avanzada. Esto, junto con el creciente interés en los últimos años de la educación universitaria basada en juegos, es la base de esta investigación. Se establecen dos objetivos: 1) ampliar el modelo epistémico de razonamiento regresivo, existente en la literatura matemática, a uno cognitivo y didáctico, y 2) establecer principios para el diseño de situaciones de enseñanza universitaria centradas en el razonamiento regresivo. Para lograr estos objetivos, se desarrollan cuatro Design experiments utilizando juegos de estrategia y problemas matemáticos. En ellos participaron un total de 322 estudiantes universitarios, desde el primer año de grado hasta el doctorado, procedentes de la Universidad Complutense de Madrid (España) y de la Università di Torino (Italia). Son estudiantes de las ramas científica y de ingeniería (Matemáticas, Ingeniería Matemática e Informática) y en la especialidad de formación de profesores (futuros profesores de matemáticas en la escuela secundaria)...Fac. de Ciencias MatemáticasTRUEunpu

Handbook of Lexical Functional Grammar

Lexical Functional Grammar (LFG) is a nontransformational theory of linguistic structure, first developed in the 1970s by Joan Bresnan and Ronald M. Kaplan, which assumes that language is best described and modeled by parallel structures representing different facets of linguistic organization and information, related by means of functional correspondences. This volume has five parts. Part I, Overview and Introduction, provides an introduction to core syntactic concepts and representations. Part II, Grammatical Phenomena, reviews LFG work on a range of grammatical phenomena or constructions. Part III, Grammatical modules and interfaces, provides an overview of LFG work on semantics, argument structure, prosody, information structure, and morphology. Part IV, Linguistic disciplines, reviews LFG work in the disciplines of historical linguistics, learnability, psycholinguistics, and second language learning. Part V, Formal and computational issues and applications, provides an overview of computational and formal properties of the theory, implementations, and computational work on parsing, translation, grammar induction, and treebanks. Part VI, Language families and regions, reviews LFG work on languages spoken in particular geographical areas or in particular language families. The final section, Comparing LFG with other linguistic theories, discusses LFG work in relation to other theoretical approaches