Slepian Wavelets for the Analysis of Incomplete Data on Manifolds

Many fields in science and engineering measure data that inherently live on non-Euclidean geometries, such as the sphere. Techniques developed in the Euclidean setting must be extended to other geometries. Due to recent interest in geometric deep learning, analogues of Euclidean techniques must also handle general manifolds or graphs. Often, data are only observed over partial regions of manifolds, and thus standard whole-manifold techniques may not yield accurate predictions. In this thesis, a new wavelet basis is designed for datasets like these. Although many definitions of spherical convolutions exist, none fully emulate the Euclidean definition. A novel spherical convolution is developed, designed to tackle the shortcomings of existing methods. The so-called sifting convolution exploits the sifting property of the Dirac delta and follows by the inner product of a function with the translated version of another. This translation operator is analogous to the Euclidean translation in harmonic space and exhibits some useful properties. In particular, the sifting convolution supports directional kernels; has an output that remains on the sphere; and is efficient to compute. The convolution is entirely generic and thus may be used with any set of basis functions. An application of the sifting convolution with a topographic map of the Earth demonstrates that it supports directional kernels to perform anisotropic filtering. Slepian wavelets are built upon the eigenfunctions of the Slepian concentration problem of the manifold - a set of bandlimited functions which are maximally concentrated within a given region. Wavelets are constructed through a tiling of the Slepian harmonic line by leveraging the existing scale-discretised framework. A straightforward denoising formalism demonstrates a boost in signal-to-noise for both a spherical and general manifold example. Whilst these wavelets were inspired by spherical datasets, like in cosmology, the wavelet construction may be utilised for manifold or graph data

Nonlinear Approximations in Filter Design and Wave Propagation

This thesis has two parts. In both parts we use nonlinear approximations to obtain accurate solutions to problems where traditional numerical approaches rapidly become computationally infeasible. The first part describes a systematic method for designing highly accurate and efficient infinite impulse response (IIR) and finite impulse response (FIR) filters given their specifications. In our approach, we first meet the specifications by constructing an IIR filter, without requiring the filter to be causal, and possibly with a large number of poles. We then construct, for any given accuracy, an optimal IIR version of such filter. Finally, also for any given accuracy, we convert the IIR filter to an efficient FIR filter cascade. In this FIR approximation, the non-causal part of the IIR filter only introduces an additional delay. Because our IIR construction does not have to enforce causality, the filters we design are more efficient than filters designed by existing methods. The second part describes a fast algorithm to propagate, for any desired accuracy, a time-harmonic electromagnetic field between two planes separated by free space. The analytic formulation of this problem (circa 1897) requires the evaluation of the Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is large, it can be accurately approximated by asymptotic methods. The computational difficulties arise in the intermediate region where, in order to obtain an accurate solution, it is necessary to apply the oscillatory Rayleigh-Sommerfeld kernel as is. In our approach, we accurately approximate the kernel by a short sum of Gaussians with complex exponents and then efficiently apply the result to input data using the unequally spaced fast Fourier transform. The resulting algorithm has the same computational complexity as methods based on the Fresnel approximation. We demonstrate that while the Fresnel approximation may provide adequate accuracy near the optical axis, the accuracy deteriorates significantly away from the optical axis. In contrast, our method maintains controlled accuracy throughout the entire computational domain

Stellar Tidal Streams as Cosmological Diagnostics: Comparing data and simulations at low galactic scales

In hierarchical models of galaxy formation, stellar tidal streams are expected around most galaxies. Although these features may provide useful diagnostics of the LCDM model, their observational properties remain poorly constrained. Statistical analysis of the counts and properties of such features is of interest for a direct comparison against results from numerical simulations. In this work, we aim to study systematically the frequency of occurrence and other observational properties of tidal features around nearby galaxies. The approach featured here is based on a visual classification of diffuse features around a sample of nearby galaxies, using a post-processing of optical survey imaging optimized for the detection of low-surfacebrightness stellar structure. At the limiting surface brightness of this sample, 14 − 17% of the galaxies exhibit evidence of diffuse features likely to have arisen from minor merging events. For simulated images, the frequency is 16 − 19%. Our technique recovers all previously known streams in the selected sample and yields a number of new candidates. We conclude that this methodology provides a reliable foundation for the statistical analysis of diffuse circumgalactic features in wide-area imaging surveys, and for the identification of targets for follow-up studies

Échantillonnage Non Uniforme : Application aux filtrages et aux conversions CAN/CNA (Convertisseurs Analogique-Numérique et Numérique/Analogique) dans les télécommunications par satellite

La théorie de l'échantillonnage uniforme des signaux, développée en particulier par C. Shannon, est à l'origine du traitement numérique du signal. Depuis, de nombreux travaux ont été consacrés à l'échantillonnage non uniforme. Celui-ci permet, d'une part, de modéliser les imperfections des dispositifs d'échantillonnage uniforme. D'autre part, l'échantillonnage peut être effectué de manière délibérément non uniforme afin de bénéficier de propriétés particulières, notamment un assouplissement des conditions portant sur le choix de la fréquence moyenne d'échantillonnage. La plupart de ces travaux reste dans un cadre théorique en adoptant des schémas d'échantillonnage et des modèles de signaux simplifiés. Or, actuellement, dans de nombreux domaines d'application, tels que les communications par satellites, la conversion analogique-numérique s'effectue sous des contraintes fortes pour les largeurs de bande mises en jeu, en raison notamment des fréquences très élevées utilisées. Ces conditions opérationnelles accentuent les imperfections des dispositifs électroniques réalisant l'échantillonnage et induisent le choix de modèles de signaux et de schémas d'échantillonnage spécifiques. Cette thèse a pour objectif général d'identifier des modèles d'échantillonnage adaptés à ce cadre applicatif. Ceux-ci s'appliquent à des signaux aléatoires passe-bande, qui constituent un modèle classique en télécommunications. Ils doivent prendre en compte des facteurs technologiques, économiques ainsi que des contraintes bord de complexité et éventuellement intégrer des fonctionnalités propres aux télécommunications. La première contribution de cette thèse est de développer des formules d'échantillonnage non uniforme qui intègrent dans le domaine numérique des fonctionnalités délicates à implémenter dans le domaine analogique aux fréquences considérées. La deuxième contribution consiste à caractériser et à compenser les erreurs de synchronisation de dispositifs d'échantillonnage non uniforme particuliers, à savoir les convertisseurs analogique-numérique entrelacés temporellement, via des méthodes supervisées ou aveugles

Optical Deformation of Microdroplets at Ultralow Interfacial Tension

What is the shape of a droplet? Its interfacial tension dictates that it is very close to a perfect sphere. Herein, the interfacial tension is reduced to ultralow values (0.1 - 100 uN/m) by careful formulation of surfactant additives, such as for mixtures that form microemulsions. The droplet need not be spherical but can accommodate external forces of a similar magnitude. The control and precision of forces afforded simply by light - in the form of highly focused Nd:YAG laser beams - are exploited in this work to deform hydrocarbon oil-in-water emulsion droplets of 1-10 um diameter. To this end, a novel, integrated platform for microfluidic generation, optical deformation and 3D fluorescent imaging of droplets is presented. Previous attempts to characterise optically-controlled microdroplet shapes have been limited to 2D projections. Here, that ambiguity is resolved using 3D confocal laser scanning- and structured illumination microscopy. 2D and 3D arrays of up to four Gaussian point traps are generated by holograms and acousto-optics. A variety of regular, prolate, oblate and asymmetric shapes are produced and correlated with parameters such as optocapillary number, trap separation and capillary length. Exotic shapes exhibiting zero or negative mean and Gaussian curvatures are presented alongside their brightfield counterparts. The complex phase behaviour of emulsion droplets and their parent phases is observed to couple strongly to thermal absorption of the beams. The rich interfacial chemistry, its relation to the forces determining droplet shape and the surprising ability to create nanofluidic networks between droplets are investigated

Galaxies with multiple bars : constraints on their formation scenarios

Much of current astrophysical research is aimed at addressing one key issue: how galaxies form and evolve. We still do not fully understand the evolutionary processes driving the lifetimes of the zoo of galaxies which populate the Universe. Galaxies may be isolated, or in groups or clusters; they may appear as pure ellipsoids or discs or include a variety of structures; they may be forming stars violently or passing quietly through their lives; and the many other properties that we are progressively discovering. Each piece we add to the puzzle pile complicates the picture a bit more. This thesis is aimed at fixing one of these pieces, a specific one related to the very interesting, but so far not well characterised, double-barred galaxies. Barred galaxies are rather common structures in the Universe and, more importantly, they are key elements for secular evolution theories. In fact, bars can transport gas to the central regions of galaxies and trigger the formation of bulges and other new structures. Double bars go a step farther: they allow the material to reach the very central regions where the gas driven inwards by a single bar cannot get to. Therefore, nested bar systems are considered a very efficient way to bring about the internal secular evolution of galaxies and even to feed active galactic nuclei. This hypothesis has, however, some detractors, as it is very promising from a theoretical point of view but there is not much observational evidence that it is valid. In this thesis we have carefully observed and studied a sample of five doublebarred galaxies using the state-of-the-art techniques for the analysis of their kinematics and stellar populations. This has led to original work not previously performed, so each result is relevant and provides clues to the nature of these objects. We are pleased to present the discovery of the σ-hollows, the only known kinematical signature of the presence of stellar inner bars. Moreover, we disentangle the complex structural composition of the galaxies of our sample, dealing with different formation scenarios and rejecting those which do not account for the observational properties derived here. We find out that one out of the five galaxies is probably hosting a disc-like bulge, whereas the other four present a classical bulge in their centres. Finally, we ascertain that inner bars tend to be younger and more metal-rich than the outer structures. Nevertheless, the characteristic stellar populations for the bulge and the inner bar are indistinguishable and there is no evidence of star-forming structures in the central regions of these galaxies, so we have to conclude that these five inner bars are not playing a major role in the secular evolution of their host galaxiesGran parte de la investigación astrofísica actual gira en torno a una única e importante cuestión: cómo se forman y evolucionan las galaxias. Todavía hoy no comprendemos los procesos evolutivos que dirigen la vida de ese zoo de objetos que puebla el Universo. Las galaxias pueden estar aisladas o viviendo en grupos o cúmulos; pueden aparentar ser simple elipsoides o discos o estar constituidas por variedad de estructuras; pueden formar estrellas violentamente o dejar pasar tranquilamente sus vidas; y un largo etcétera de propiedades que vamos descubriendo poco a poco. Cada pieza que añadimos al puzle complica la imagen final un poco más. Esta tesis pretende colocar por fin una de esas piezas, aquella relacionada con el fascinante, aunque todavía poco estudiado, caso de las galaxias con dos barras. Las galaxias con barra son estructuras comunes en el Universo y elementos clave en las teorías de evolución secular, puesto que las barras son capaces de transportar gas a las regiones centrales y, por lo tanto, promover la formación de bulbos y otras estructuras. Las dobles barras van un paso más allá al permitir que el material alcance las partes más internas de las galaxias, donde el gas transportado por una barra simple no puede llegar. Es por tanto lógico que estos sistemas dobles sean considerados fundamentales y muy eficientes para la evolución secular interna de las galaxias, e incluso se les relaciona a menudo con los núcleos activos galácticos. Esta hipótesis encuentra, sin embargo, varios detractores, pues aunque es muy prometedora desde un punto de vista puramente teórico, no hay evidencias observacionales que la sustenten. En la presente tesis hemos observado y estudiado en detalle una muestra de cinco galaxias con dos barras y hemos hecho uso de las técnicas de análisis más novedosas con el objetivo de caracterizar su cinemática y poblaciones estelares. Se trata de un trabajo completamente original jamás antes realizado, por lo que cada resultado es relevante y proporciona nuevas pistas sobre la naturaleza de estos objetos. Así pues, presentamos el descubrimiento de los llamados σ-hollows, las únicas señales cinemáticas conocidas de la presencia de barras internas. Desentrañamos también la compleja composición estructural de las galaxias de nuestra muestra, considerando los diferentes escenarios de formación posibles y rechazando aquellos que no explican las propiedades observacionales encontradas. Obtenemos así que una de las cinco galaxias en estudio probablemente contiene un bulbo tipo disco, mientras que el resto de la muestra está compuesta por bulbos clásicos. Finalmente, averiguamos que las barras internas tienden a ser más jóvenes y metálicas que las estructuras externas; sin embargo, las poblaciones estelares características para el bulbo y la barra interna son indistinguibles y no hay evidencias de formación estelar relevante en las regiones centrales, por lo que concluimos que al menos estas cinco barras internas no están desempeñando un papel importante en la evolución secular de sus galaxias anfitrionas

Analysis of Images of Comet 67P/Churyumov-Gerasimenko Obtained by Osiris/Rosetta: Dust Particles Investigation

The aim of this thesis is to study and characterize the dust in the coma of the comet 67P/Churyumov-Gerasimenko in order to increase our knowledge about the nature and behavior of comets. These objects are the most pristine of our solar system and their investigation allows to understand the physical and chemical conditions that took place in the primordial environment where they formed. In turn, dust is a fundamental constituent of comets and provides valuable information on their composition, structure and evolution. The importance of the dust lies in the possibility to use it as a proxy to derive properties of the interior of the cometary nuclei. Indeed, the activity produced when the comet approaches the Sun releases both surface material and pristine matter from the inside of the nucleus that cannot be reached otherwise. The ESA Rosetta mission to the comet 67P/Churyumov-Gerasimenko revealed a new insight of the cometary dust providing precious information about its composition, dimension, size distribution and structure. For the rst time it was possible to acquire images of single dust aggregates and to give a comprehensive description of their properties, thanks to the information obtained by all the instrumentation on board the spacecraft. The overall processes that act on the comets are not yet fully understood and there are still open questions that need to be claried. It is therefore essential to continue to investigate this complex scenario to unveil the real nature of these primordial objects. The rst work of this thesis concerns the photometric analysis of images of dust grains, taken by the OSIRIS Narrow Angle Camera (NAC). The large amount of data forced us to develop an automatic pipeline to identify and to analyze the particle tracks present in the images. We measured the spectral slope of about 2000 grains in 555 images taken in 18 days of observation from July 2015 to January 2016. Then we evaluate their variability with respect to the distance from the nucleus and from the Sun. To better characterize the grains composition we perform also a spectrophotometric analysis of a subgroup of 339 grains. To check their nature, we compared their three point spectra with the ones of several nucleus terrains identifying dierent types of grains spectra. The second work of this thesis treats the scattering properties of cometary dust analogs, in collaboration with the Instituto de Astrofsica de Andalusa, Granada at the Cosmic Dust Laboratory. Laboratory experiments are of fundamental importance to interpret the data since they allow to study complex phenomena in a monitored environment, favoring the evaluation of the eects due to specic physical and chemical properties of the dust. We measure the phase function and the linear polarization curve of 7 samples, consisting on 4 meteorites, 2 silicates and 1 organic compound with the aim to compare our results with the observational data of 67P and other comets. Our last work, which is still in progress, consists in the analysis of dust motion with the aim to provide observational constraints to the dynamical models of cometary dust in the inner coma as well as to the shape of the single grains. We focus our investigation on the 15 groups of images taken from July 2015 to January 2016 and we measure the rotational periods, the axis ratio and the direction of motion on a dataset of more than 500 grains