Visualization, processing and editing of audio, directly on a time-frequency surface, is the scope of this thesis. More precisely the scalogram produced by a Gabor Wavelet transform is used, which is a powerful alternative to traditional techinques where the wave form is the main visual aid and editting is performed by parametric filters. Reconstruction properties, scalogram design and enhancements as well audio manipulation algorithms are investigated for this audio representation.The scalogram is designed to allow a flexible choice of time-frequency ratio, while maintaining high quality reconstruction. For this mean, the Loglet is used, which is observed to be the most suitable filter choice.  Re-assignmentare tested, and a novel weighting function using partial derivatives of phase is proposed.  An audio interpolation procedure is developed and shown to perform well in listening tests.The feasibility to use the transform coefficients directly for various purposes is investigated. It is concluded that Pitch shifts are hard to describe in the framework while noise thresh holding works well. A downsampling scheme is suggested that saves on operations and memory consumption as well as it speeds up real world implementations significantly. Finally, a Scalogram 'compression' procedure is developed, allowing the caching of an approximate scalogram.

In this paper we deal with the `strength' of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according_to_M. Z. Nashed [15] the ill-posedness of type I if A is not compact, but we have R(A) 6= R(A) for the range R(A) of A; and the ill-posedness of type II for compact operators A: From our considerations it seems to follow that the problems with noncompact operators A are not in general `less' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed in [10] are derived from the decay rate of the nonincreasing sequence of singular values of A. Since singular values do not exist for noncompact operators A; we introduce stability rates in order to have a common measure for the compact and noncompact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the noncompact case. Moreover using increasing rearrangements of the multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for the multiplication operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.

Diese Dissertation ist dem Studium temperierter Distributionen mittels mikrolokaler Methoden gewidmet. Die fundamentale Größe der mikrolokalen Analysis, die Wellenfrontmenge, wird durch zwei analoge Konzepte ersetzt, die den pseudo-differentiellen SG- und Shubin-Kalkülen zugeordnet sind. Die Eigenschaften dieser globalen Wellenfrontmengen werden studiert und ferner werden unterschiedliche Möglichkeiten, diese globalen Singularitäten zu charakterisieren, untersucht, insbesondere mittels der FBI-Transformation. Zahlreiche Konstruktionen, die den klassischen Wellenfrontmengenbegriff beinhalten, werden in den globalen Kontext übersetzt, insbesondere Rechenoperationen mit temperierten Distributionen wie etwa (getwistete) Produkte, Pull-backs und Paarungen, für die mikrolokale Existenzkriterien angegeben werden. Als eine Anwendung wird eine Klasse von temperierten Oszillatorintegralen eingeführt, welche durch inhomogene Phasenfunktionen und Amplituden aus SG-Symbolklassen parametrisiert werden. Die SG-Wellenfrontmengen dieser Distributionen werden untersucht und es stellt sich heraus, dass diese durch eine Verallgemeinerung der Menge stationärer Punkte der Phasenfunktionen beschränkt werden. In diesem Kontext wird eine Verallgemeinerung des klassischen Begriffs einer konischen Lagrange-Untermannifaltigkeit des T*R^d vorgenommen und diese Objekte werden auf ihre Parametrisierungseigenschaften untersucht. Es stellt sich heraus, dass jedes solche Objekt lokal als die Menge der stationären Punkte einer SG-Phasenfunktion realisiert werden kann. Als weitere Anwendung werden einige Konstruktionen der axiomatischen Quantenfeldtheorie, die Distributionen beinhalten, im temperierten Kontext realisiert.

Cette thèse se propose d'obtenir des résultats statistiques sur les zéros non-triviaux de fonctions L. Dans le cas des fonctions L de formes modulaires, on prouve qu'une proportion positive explicite de zéros non-triviaux se situe sur la droite critique. Afin d'arriver à ce résultat, il nous faut préalablement étendre un théorème sur les problèmes de convolution avec décalage additif en moyenne de manière à déterminer le comportement asymptotique du second moment intégral ramolli d'une fonction L de forme modulaire au voisinage de la droite critique. Une autre partie de cette thèse, indépendante de la précédente, est consacrée à l'étude du plus petit zéro non-trivial d'une famille de fonctions L. Ces résultats sont en particulier appliqués aux fonctions L de puissance symétrique.

This dissertation is dedicated to study the design and utilization of financial contracts and pricing mechanisms for managing the demand/price risks in electricity markets and the price risks in carbon emission markets from different perspectives. We address the issues pertaining to the efficient computational algorithms for pricing complex financial options which include many structured energy financial contracts and the design of economic mechanisms for managing the risks associated with increasing penetration of renewable energy resources and with trading emission allowance permits in the restructured electric power industry. To address the computational challenges arising from pricing exotic energy derivatives designed for various hedging purposes in electricity markets, we develop a generic computational framework based on a fast transform method, which attains asymptotically optimal computational complexity and exponential convergence. For the purpose of absorbing the variability and uncertainties of renewable energy resources in a smart grid, we propose an incentive-based contract design for thermostatically controlled loads (TCLs) to encourage end users' participation as a source of DR. Finally, we propose a market-based approach to mitigate the emission permit price risks faced by generation companies in a cap-and-trade system. Through a stylized economic model, we illustrate that the trading of properly designed financial options on emission permits reduces permit price volatility and the total emission reduction cost.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T18:45:49Z No. of bitstreams: 1 HILDÊNIO JOSÉ MACEDO - DISSERTAÇÃO PPGMAT 2011..pdf: 310824 bytes, checksum: ce96943d42ca2ee474b2fd99f6612b5c (MD5) / Made available in DSpace on 2018-07-25T18:45:49Z (GMT). No. of bitstreams: 1 HILDÊNIO JOSÉ MACEDO - DISSERTAÇÃO PPGMAT 2011..pdf: 310824 bytes, checksum: ce96943d42ca2ee474b2fd99f6612b5c (MD5) Previous issue date: 2011-05 / CNPq / Na presente dissertação, estudamos a existência e unicidade de solução para o problema de Cauchy associado a equação de evolução não local (Baixar arquivo para ver a equação). Exibimos um funcional energia, associado a esta equação, e verificamos que ele satisfaz a propriedade de Lyapunov. Além disso, usamos este funcional para mostrar a existência e estabilidade local de uma solução de equilíbrio referida na literatura como instanton. / In this work we prove existence and uniqueness of solution for the Cauchy problem corresponding to nonlocal evolution equation (Download file to see the equation). We exhibit an energy functional associated to this equation, and verify that it satisfies the Lyapunov property. Moreover, use this function to show the existence and local stability of a equilibrium solution reported in the literature as instanton.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-05T12:56:54Z No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) / Approved for entry into archive by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-05T15:29:04Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) / Made available in DSpace on 2017-09-05T15:29:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) Previous issue date: 2016-08-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We will do here a theoretical study of the Discrete Fourier Transform on the finite circle ℤ/nℤ. Our main objective is to see if we can get properties analogous to those found in the Fourier transform for the continuous case. In this work we show that ℤ/nℤ has a ring structure, providing conditions for the development of extensively discussed topics in arithmetic, for example, The Chinese Remainder Theorem, Euler’s Phi Function and primitive roots, themes these to be dealt with in first chapter. The main subject of this study is developed in the second chapter, which define the space L2(ℤ/nℤ) and prove that this is a finite-dimensional inner product vector space, with an orthonormal basis. This fact is of utmost importance when we are determining the matrix and demonstrating the properties of the discrete Fourier transform. We will also make geometric interpretations of the Chinese Remainder Theorem and the finite circle ℤ/nℤ as well as give a graphical representation of the DFT of some functions that calculate. During the development of this study we will make recurrent use of definitions and results treated in Arithmetic, Algebra and Linear Algebra. / Faremos, aqui, um estudo teórico sobre a Transformada Discreta de Fourier no círculo finito ℤ/nℤ. Nosso principal objetivo é verificar se podemos obter propriedades análogas às encontradas nas transformadas de Fourier para o caso contínuo. Nesse trabalho mostraremos que ℤ/nℤ tem uma estrutura de anel, dando condições para o desenvolvimento de temas bastante discutidos na Aritmética como, por exemplo, o Teorema Chinês do Resto, função Phi de Euler e raízes primitivas, temas estes que serão tratados no primeiro capítulo. O assunto principal desse estudo é desenvolvido no segundo capítulo, onde definiremos o espaço L2(ℤ/nℤ) e provaremos que este é um espaço vetorial com produto interno, dimensão finita e uma base ortonormal. Tal fato será de extrema importância quando estivermos determinando a matriz e demonstrando as propriedades da transformada discreta de Fourier. Também faremos interpretações geométricas do Teorema Chinês do Resto e do círculo finito ℤ/nℤ assim como daremos a representação gráfica da DFT de algumas funções que calcularemos. Durante o desenvolvimento desse estudo faremos uso recorrente de definições e resultados tratados na Aritmética, Álgebra e Álgebra Linear.

O objetivo deste trabalho é estudar aproximação na esfera por uma soma com pesos de harmônicos esféricos. Apresentamos condições necessárias e suficientes sobre os pesos para garantir a convergência, tanto no caso contínuo quanto no caso Lp. Analisamos a ordem de convergência dos processos aproximatórios usando um módulo de suavidade esférico relacionado à derivada forte de Laplace-Beltrami. Incluímos provas para vários resultados sobre a derivada forte de Laplace-Beltrami, já que não conseguimos encontrá-las na literatura / The subject of this work is to study approximation on the sphere by weighted sums of spherical harmonics. We present necessary and sufficient conditions on the weights for convergence in both, the continuous and the Lp cases. We analyse the convergence rates of the approximation processes using a modulus of smoothness related to the strong Laplace- Beltrami derivative. We include proofs for several results related to such a derivative, since we were unable to find them in the literature

Abstract: Nowadays, traffic analysis are relying on data collected from various traffic sensors. Among the various traffic surveillance techniques, video surveillance systems are often used for monitoring and characterizing traffic load. In this thesis, we focused on two aspects of traffic analysis without using motion features in low frame-rate videos: Traffic density flow analysis and Vehicle detection and classification. Traffic density flow analysis}: Knowing in real time when the traffic is fluid or when it jams is a key information to help authorities re-route vehicles and reduce congestion. Accurate and timely traffic flow information is strongly needed by individual travelers, the business sectors and government agencies. In this part, we investigated the possibility of monitoring highway traffic based on videos whose frame rate is too low to accurately estimate motion features. As we are focusing on analyzing traffic images and low frame-rate videos, traffic density is defined as the percentage of road being occupied by vehicles. In our previous work, we validated that traffic status is highly correlated to its texture features and that Convolutional Neural Networks (CNN) has the superiority of extracting discriminative texture features. We proposed several CNN models to segment traffic images into three different classes (road, car and background), classify traffic images into different categories (empty, fluid, heavy, jam) and predict traffic density without using any motion features. In order to generalize the model trained on a specific dataset to analyze new traffic scenes, we also proposed a novel transfer learning framework to do model adaptation. Vehicle detection and classification: The detection of vehicles pictured by traffic cameras is often the very first step of video surveillance systems, such as vehicle counting, tracking and retrieval. In this part, we explore different deep learning methods applied to vehicle detection and classification. Firstly, realizing the importance of large dataset for traffic analysis, we built and released the largest traffic dataset (MIO-TCD) in the world for vehicle localization and classification in collaboration with colleagues from Miovision inc. (Waterloo, On). With this dataset, we organized the Traffic Surveillance Workshop and Challenge in conjunction with CVPR 2017. Secondly, we evaluated several state-of-the-art deep learning methods for the classification and localization task on the MIO-TCD dataset. In light of the results, we may conclude that state-of-the-art deep learning methods exhibit a capacity to localize and recognize vehicle from a single video frame. While with a deep analysis of the results, we also identify scenarios for which state-of-the-art methods are still failing and propose concrete ideas for future work. Lastly, as saliency detection aims to highlight the most relevant objects in an image (e.g. vehicles in traffic scenes), we proposed a multi-resolution 4*5 grid CNN model for the salient object detection. The model enables near real-time high performance saliency detection. We also extend this model to do traffic analysis, experiment results show that our model can precisely segment foreground vehicles in traffic scenes. / De nos jours, l’analyse de trafic routier est de plus en plus automatisée et s’appuie sur des données issues de senseurs en tout genre. Parmi les approches d’analyse de trafic routier figurent les méthodes à base de vidéo. Les méthodes à base de vidéo ont pour but d’identifier et de reconnaître les objets en mouvement (généralement des voitures et des piétons) et de comprendre leur dynamique. Un des défis parmi les plus difficile à résoudre est d’analyser des séquences vidéo dont le nombre d’images par seconde est très faible. Ce type de situation est pourtant fréquent considérant qu’il est très difficile (voir impossible) de transmettre et de stocker sur un serveur un très grand nombre d’images issues de plusieurs caméras. Dans ce cas, les méthodes issues de l’état de l’art échouent car un faible nombre d’images par seconde ne permet pas d’extraire les caractéristiques vidéos utilisées par ces méthodes tels le flux optique, la détection de mouvement et le suivi de véhicules. Au cours de cette thèse, nous nous sommes concentré sur l’analyse de trafic routier à partir de séquences vidéo contenant un très faible nombre d’images par seconde. Plus particulièrement, nous nous sommes concentrés sur les problème d’estimation de la densité du trafic routier et de la classification de véhicules. Pour ce faire, nous avons proposé différents modèles à base de réseaux de neurones profonds (plus particulièrement des réseaux à convolution) ainsi que de nouvelles bases de données permettant d’entraîner les dits modèles. Parmi ces bases de données figure « MIO-TCD », la plus grosse base de données annotées au monde faite pour l’analyse de trafic routier.

The focus of this thesis is variational image restoration techniques that involve novel non-smooth first-order gradient regularisers: Total Variation (TV) regularisation in image and data space for reconstruction of thin structures from PET data and regularisers given by an infimal-convolution of TV and $L^p$ seminorms for denoising images with piecewise affine structures. In the first part of this thesis, we present a novel variational model for PET reconstruction. During a PET scan, we encounter two different spaces: the sinogram space that consists of all the PET data collected from the detectors and the image space where the reconstruction of the unknown density is finally obtained. Unlike most of the state of the art reconstruction methods in which an appropriate regulariser is designed in the image space only, we introduce a new variational method incorporating regularisation in image and sinogram space. In particular, the corresponding minimisation problem is formed by a total variational regularisation on both the sinogram and the image and with a suitable weighted $L^2$ fidelity term, which serves as an approximation to the Poisson noise model for PET. We establish the well-posedness of this new model for functions of Bounded Variation (BV) and perform an error analysis through the notion of the Bregman distance. We examine analytically how TV regularisation on the sinogram affects the reconstructed image especially the boundaries of objects in the image. This analysis motivates the use of a combined regularisation principally for reconstructing images with thin structures. In the second part of this thesis we propose a first-order regulariser that is a combination of the total variation and $L^p$ seminorms with $1 < p \le \infty$. A well-posedness analysis is presented and a detailed study of the one dimensional model is performed by computing exact solutions for simple functions such as the step function and a piecewise affine function, for the regulariser with $p = 2$ and $p = 1$. We derive necessary and sufficient conditions for a pair in $BV \times L^p$ to be a solution for our proposed model and determine the structure of solutions dependent on the value of $p$. In the case $p = 2$, we show that the regulariser is equivalent to the Huber-type variant of total variation regularisation. Moreover, there is a certain class of one dimensional data functions for which the regularised solutions are equivalent to high-order regularisers such as the state of the art total generalised variation (TGV) model. The key assets of our regulariser are the elimination of the staircasing effect - a well-known disadvantage of total variation regularisation - the capability of obtaining piecewise affine structures for $p = 1$ and qualitatively comparable results to TGV. In addition, our first-order $TVL^p$ regulariser is capable of preserving spike-like structures that TGV is forced to smooth. The numerical solution of the proposed first-order model is in general computationally more efficient compared to high-order approaches.