Muitos métodos de redução de ruído se baseiam na possibilidade de representar o sinal original com um reduzido número de coeficientes de uma transformada, ou melhor, obtém-se um sinal com menos ruído pelo cancelamento dos coeficientes abaixo de um valor adequadamente estabelecido de magnitude. Deve-se supor que a contribuição do ruído se distribua de maneira uniforme por todos os coeficientes. Uma desvantagem destes métodos, quando aplicados a sinais de voz, é a distorção introduzida pela eliminação dos coeficientes de pequena magnitude, juntamente com a presença de sinais espúrios, como o “ruído musical" produzido por coeficientes ruidosos isolados que eventualmente ultrapassam o limiar. Para as transformadas usualmente empregadas, o histograma da distribuição dos coeficientes do sinal de voz possui um grande número de coeficientes próximos à origem. Diante disto, propomos uma nova função de “thresholding" concebida especialmente para redução de ruído em sinais de voz adicionados a AWGN (“Additive, White, and Gaussian Noise"). Esta função, chamada de SoftSoft, depende de dois valores de limiar: um nível inferior, ajustado para reduzir a distorção da voz, e um nível superior, ajustado para eliminar ruído. Os valores ótimos de limiar são calculados para minimizar uma estimativa do erro quadrático médio (MSE): diretamente, supondo conhecido o sinal original; indiretamente, usando uma função de interpolação para o MSE, levando a um método prático. A função SoftSoft alcança um MSE inferior ao que se obtém pelo emprego das conhecidas operações de “Soft" ou “Hard-thresholding", as quais dispõem apenas do limiar superior. Ainda que a melhoria em termos de MSE não seja muito expressiva, a melhoria da qualidade perceptual foi certificada tanto por um ouvinte quanto por uma medida perceptual de distorção (a distância log-espectral). / Many noise-reduction methods are based on the possibility of representing the clean signal as a reduced number of coefficients of a block transform, so that cancelling coefficients below a certain thresholding level will produce an enhanced reconstructed signal. It is necessary to assume that the clean signal has a sparse representation, while the noise energy is spread over all coefficients. The main drawback of those methods is the speech distortion introduced by eliminating small magnitude coefficients, and the presence of artifacts (“musical noise") produced by isolated noisy coefficients randomly crossing the thresholding level. Based on the observation that the speech coefficient histogram has many important coefficients close to origin, we propose a custom thresholding function to perform noise reduction in speech signals corrupted by AWGN. This function, called SoftSoft, has two thresholding levels: a lower level adjusted to reduce speech distortion, and a higher level adjusted to remove noise. The joint optimal values can be determined by minimizing the resulting mean square error (MSE). We also verify that this new thresholding function leads to a lower MSE than the well-known Soft and Hard-thresholding functions, which employ only a higher thresholding level. Although the improvement in terms of MSE is not expressive, a perceptual distortion measure (the log-spectral distance, LSD) is employed to prove the higher performance of the proposed thresholding scheme.

Muitos métodos de redução de ruído se baseiam na possibilidade de representar o sinal original com um reduzido número de coeficientes de uma transformada, ou melhor, obtém-se um sinal com menos ruído pelo cancelamento dos coeficientes abaixo de um valor adequadamente estabelecido de magnitude. Deve-se supor que a contribuição do ruído se distribua de maneira uniforme por todos os coeficientes. Uma desvantagem destes métodos, quando aplicados a sinais de voz, é a distorção introduzida pela eliminação dos coeficientes de pequena magnitude, juntamente com a presença de sinais espúrios, como o “ruído musical” produzido por coeficientes ruidosos isolados que eventualmente ultrapassam o limiar. Para as transformadas usualmente empregadas, o histograma da distribuição dos coeficientes do sinal de voz possui um grande número de coeficientes próximos à origem. Diante disto, propomos uma nova função de “thresholding” concebida especialmente para redução de ruído em sinais de voz adicionados a AWGN (“Additive, White, and Gaussian Noise”). Esta função, chamada de SoftSoft, depende de dois valores de limiar: um nível inferior, ajustado para reduzir a distorção da voz, e um nível superior, ajustado para eliminar ruído. Os valores ótimos de limiar são calculados para minimizar uma estimativa do erro quadrático médio (MSE): diretamente, supondo conhecido o sinal original; indiretamente, usando uma função de interpolação para o MSE, levando a um método prático. A função SoftSoft alcança um MSE inferior ao que se obtém pelo emprego das conhecidas operações de “Soft” ou “Hard-thresholding”, as quais dispõem apenas do limiar superior. Ainda que a melhoria em termos de MSE não seja muito expressiva, a melhoria da qualidade perceptual foi certificada tanto por um ouvinte quanto por uma medida perceptual de distorção (a distância log-espectral). / Many noise-reduction methods are based on the possibility of representing the clean signal as a reduced number of coefficients of a block transform, so that cancelling coefficients below a certain thresholding level will produce an enhanced reconstructed signal. It is necessary to assume that the clean signal has a sparse representation, while the noise energy is spread over all coefficients. The main drawback of those methods is the speech distortion introduced by eliminating small magnitude coefficients, and the presence of artifacts (“musical noise”) produced by isolated noisy coefficients randomly crossing the thresholding level. Based on the observation that the speech coefficient histogram has many important coefficients close to origin, we propose a custom thresholding function to perform noise reduction in speech signals corrupted by AWGN. This function, called SoftSoft, has two thresholding levels: a lower level adjusted to reduce speech distortion, and a higher level adjusted to remove noise. The joint optimal values can be determined by minimizing the resulting mean square error (MSE). We also verify that this new thresholding function leads to a lower MSE than the well-known Soft and Hard-thresholding functions, which employ only a higher thresholding level. Although the improvement in terms of MSE is not expressive, a perceptual distortion measure (the log-spectral distance, LSD) is employed to prove the higher performance of the proposed thresholding scheme.

Accurate and fast machine learning (ML) algorithms are highly vital in artificial intelligence (AI) applications. In complex dataset problems, traditional ML methods such as radial basis function neural network (RBFN), sparse coding (SC) using dictionary learning, and particle swarm optimization (PSO) provide trivial results, large structure, slow training, and/or slow testing. This dissertation introduces four novel ML techniques: the multi-column RBFN network (MCRN), the projected dictionary learning algorithm (PDL) and the multi-column adaptive and non-adaptive particle swarm optimization techniques (MC-APSO and MC-PSO). These novel techniques provide efficient alternatives for traditional ML techniques. Compared to traditional ML techniques, the novel ML techniques demonstrate more accurate results, faster training and testing timing, and parallelized structured solutions. MCRN deploys small RBFNs in a parallel structure to speed up both training and testing. Each RBFN is trained with a subset of the dataset and the overall structure provides results that are more accurate. PDL introduces a conceptual dictionary learning method in updating the dictionary atoms with the reconstructed input blocks. This method improves the sparsity of extracted features and hence, the image denoising results. MC-PSO and MC-APSO provide fast and more accurate alternatives to the PSO and APSO slow evolutionary techniques. MC-PSO and MC-APSO use multi-column parallelized RBFN structure to improve results and speed with a wide range of classification dataset problems. The novel techniques are trained and tested using benchmark dataset problems and the results are compared with the state-of-the-art counterpart techniques to evaluate their performance. Novel techniques’ results show superiority over techniques in accuracy and speed in most of the experimental results, which make them good alternatives in solving difficult ML problems.

Natural human motion is highly demanded and widely used in a variety of applications such as video games and virtual realities. However, acquisition of full-body motion remains challenging because the system must be capable of accurately capturing a wide variety of human actions and does not require a considerable amount of time and skill to assemble. For instance, commercial optical motion capture systems such as Vicon can capture human motion with high accuracy and resolution while they often require post-processing by experts, which is time-consuming and costly. Microsoft Kinect, despite its high popularity and wide applications, does not provide accurate reconstruction of complex movements when significant occlusions occur. This dissertation explores two different approaches that accurately reconstruct full-body human motion from noisy and ambiguous input data captured by commercial motion capture devices. The first approach automatically generates high-quality human motion from noisy data obtained from commercial optical motion capture systems, eliminating the need for post-processing. The second approach accurately captures a wide variety of human motion even under significant occlusions by using color/depth data captured by a single Kinect camera. The common theme that underlies two approaches is the use of prior knowledge embedded in pre-recorded motion capture database to reduce the reconstruction ambiguity caused by noisy and ambiguous input and constrain the solution to lie in the natural motion space. More specifically, the first approach constructs a series of spatial-temporal filter bases from pre-captured human motion data and employs them along with robust statistics techniques to filter noisy motion data corrupted by noise/outliers. The second approach formulates the problem in a Maximum a Posterior (MAP) framework and generates the most likely pose which explains the observations as well as consistent with the patterns embedded in the pre-recorded motion capture database. We demonstrate the effectiveness of our approaches through extensive numerical evaluations on synthetic data and comparisons against results created by commercial motion capture systems. The first approach can effectively denoise a wide variety of noisy motion data, including walking, running, jumping and swimming while the second approach is shown to be capable of accurately reconstructing a wider range of motions compared with Microsoft Kinect.

To handle potentially large size and complicated nonstationary functional data, we present the wavelet-based methodology in data mining for process monitoring and fault classification. Since traditional wavelet shrinkage methods for data de-noising are ineffective for the more demanding data reduction goals, this thesis presents data reduction methods based on discrete wavelet transform. Our new methods minimize objective functions to balance the tradeoff between data reduction and modeling accuracy. Several evaluation studies with four popular testing curves used in the literature and with two real-life data sets demonstrate the superiority of the proposed methods to engineering data compression and statistical data de-noising methods that are currently used to achieve data reduction goals. Further experimentation in applying a classification tree-based data mining procedure to the reduced-size data to identify process fault classes also demonstrates the excellence of the proposed methods. In this application the proposed methods, compared with analysis of original large-size data, result in lower misclassification rates with much better computational efficiency. This thesis extends the scalogram's ability for handling noisy and possibly massive data which show time-shifted patterns. The proposed thresholded scalogram is built on the fast wavelet transform, which can effectively and efficiently capture non-stationary changes in data patterns. Finally, we present a SPC procedure that adaptively determines which wavelet coefficients will be monitored, based on their shift information, which is estimated from process data. By adaptively monitoring the process, we can improve the performance of the control charts for functional data. Using a simulation study, we compare the performance of some of the recommended approaches.

Recently, solutions to the problem of image segmentation and denoising are developed based on the Mumford-Shah model. The model provides an energy functional, called the Mumford-Shah functional, which should be minimized. Since the minimization of the functional has some difficulties, approximate approaches are proposed. Two such methods are the gradient flows method and the Chan-Vese active contour method. The performance evolution in terms of speed shows that the gradient flows method converges to the boundaries of the smooth parts faster / but for the hierarchical four-phase segmentation, it is observed that this method sometimes gives unsatisfactory results. In this work, a fast hierarchical four-phase segmentation method is proposed where the Chan-Vese active contour method is applied following the gradient flows method. After the segmentation process, the segmented regions are denoised using diffusion filters. Additionally, for the low signal-to-noise ratio applications, the prefiltering scheme using nonlinear diffusion filters is included in the proposed method. Simulations have shown that the proposed method provides an effective solution to the image segmentation and denoising problem.

Many mathematical models have been designed to remove noise from images. Most of them focus on grey value images with additive artificial noise. Only very few specifically target natural color photos taken by a digital camera with real noise. Noise in natural color photos have special characteristics that are substantially different from those that have been added artificially. In this thesis previous denoising models are reviewed. We analyze the strengths and weakness of existing denoising models by showing where they perform well and where they don't. We put special focus on two models: The steering kernel regression model and the non-local model. For Kernel Regression model, an adaptive bilateral filter is introduced as complementary to enhance it. Also a non-local bilateral filter is proposed as an application of the idea of non-local means filter. Then the idea of cross-channel denoising is proposed in this thesis. It is effective in denoising monochromatic images by understanding the characteristics of digital noise in natural color images. A non-traditional color space is also introduced specifically for this purpose. The cross-channel paradigm can be applied to most of the exisiting models to greatly improve their performance for denoising natural color images.

In a series of work initiated by Nemirovsky and Yudin, and later extended by Nesterov, first-order algorithms for unconstrained minimization with optimal theoretical complexity bound have been proposed. On the other hand, conjugate gradient algorithms as one of the widely used first-order techniques suffer from the lack of a finite complexity bound. In fact their performance can possibly be quite poor. This dissertation is partially on tightening the gap between these two classes of algorithms, namely the traditional conjugate gradient methods and optimal first-order techniques. We derive conditions under which conjugate gradient methods attain the same complexity bound as in Nemirovsky-Yudin's and Nesterov's methods. Moreover, we propose a conjugate gradient-type algorithm named CGSO, for Conjugate Gradient with Subspace Optimization, achieving the optimal complexity bound with the payoff of a little extra computational cost. We extend the theory of CGSO to convex problems with linear constraints. In particular we focus on solving $l_1$-regularized least square problem, often referred to as Basis Pursuit Denoising (BPDN) problem in the optimization community. BPDN arises in many practical fields including sparse signal recovery, machine learning, and statistics. Solving BPDN is fairly challenging because the size of the involved signals can be quite large; therefore first order methods are of particular interest for these problems. We propose a quasi-Newton proximal method for solving BPDN. Our numerical results suggest that our technique is computationally effective, and can compete favourably with the other state-of-the-art solvers.

This thesis presents a novel set of image analysis tools developed for the purpose of assisting radiologists with the task of detecting and characterizing breast lesions in image data acquired using magnetic resonance imaging (MRI). MRI is increasingly being used in the clinical setting as an adjunct to x-ray mammography (which is, itself, the basis of breast cancer screening programs worldwide) and ultrasound. Of these imaging modalities, MRI has the highest sensitivity to invasive cancer and to multifocal disease. MRI is the most reliable method for assessing tumour size and extent compared to the gold standard histopathology. It also shows great promise for the improved screening of younger women (with denser, more radio opaque breasts) and, potentially, for women at high risk. Breast MRI presently has two major shortcomings. First, although its sensitivity is high its specificity is relatively poor; i.e. the method detects many false positives. Second, the method involves acquiring several high-resolution image volumes before, during and after the injection of a contrast agent. The large volume of data makes the task of interpretation by the radiologist both complex and time-consuming. These shortcomings have motivated the research and development of the computer-aided detection systems designed to improve the efficiency and accuracy of interpretation by the radiologist. Whilst such systems have helped to improve the sensitivity/specificity of interpretation, it is the premise of this thesis that further gains are possible through automated image analysis. However, the automated analysis of breast MRI presents several technical challenges. This thesis investigates several of these, noise filtering, parametric modelling of contrast enhancement, segmentation of suspicious tissue and quantitative characterisation and classification of suspicious lesions. In relation to noise filtering, a new denoising algorithm for dynamic contrast-enhanced (DCE-MRI) data is presented, called the Dynamic Non-Local Means (DNLM). The DCE-MR image data is inherently contaminated by Rician noise and, additionally, the limited acquisition time per volume and the use of fat-suppression diminishes the signal-to-noise ratio. The DNLM algorithm, specifically designed for the DCE-MRI, is able to attenuate this noise by exploiting the redundancy of the information between the different temporal volumes, while taking into account the contrast enhancement of the tissue. Empirical results show that the algorithm more effectively attenuates noise in the DCE-MRI data than any of the previously proposed algorithms. In relation to parametric modelling of contrast enhancement, a new empiric model of contrast enhancement has been developed that is parsimonious in form. The proposed model serves as the basis for the segmentation and feature extraction algorithms presented in the thesis. In contrast to pharmacokinetic models, the proposed model does not rely on measured parameters or constants relating to the type or density of the tissue. It also does not assume a particular relationship between the observed changes in signal intensity and the concentration of the contrast agent. Empirical results demonstrate that the proposed model fits real data better than either the Tofts or Brix models and equally as well as the more complicated Hayton model. In relation to the automatic segmentation of suspicious lesions, a novel method is presented, based on seeded region growing and merging, using criteria based on both the original image MR values and the fitted parameters of the proposed model of contrast enhancement. Empirical results demonstrate the efficacy of the method, both as a tool to assist the clinician with the task of locating suspicious tissue and for extracting quantitative features. Finally, in relation to the quantitative characterisation and classification of suspicious lesions, a novel classifier (i.e. a set of features together with a classification method) is presented. Features were extracted from noise-filtered and segmented-image volumes and were based both on well-known features and several new ones (principally, on the proposed model of contrast enhancement). Empirical results, based on routine clinical breast MRI data, show that the resulting classifier performs better than other such classifiers reported in the literature. Therefore, this thesis demonstrates that improvements in both sensitivity and specificity are possible through automated image analysis.

Le sujet principal de cette thèse est de proposer de nouvelles méthodes d'extractions de signaux avec applications en finance. Par signaux, nous entendons soit un signal sur lequel repose une stratégie d'investissement; soit un signal perturbé par un bruit, que nous souhaitons retrouver. Ainsi, la première partie de la thèse étudie la contagion en volatilité historique autours des annonces de résultats des entreprises du Nasdaq. Nous trouvons qu'autours de l'annonce, l'entreprise reportant ses résultats, génère une contagion persistante en volatilité à l’encontre des entreprises appartenant au même secteur. Par ailleurs, nous trouvons que la contagion en volatilité varie, selon le type de nouvelles reportées, l'effet de surprise, ou encore par le sentiment de marché à l'égard de l'annonceur. La deuxième partie de cette thèse adapte des techniques de dé-bruitage venant de l'imagerie, à des formes de bruits présentent en finance. Ainsi, un premier article, co-écrit avec Matthieu Garcin, propose une technique de dé-bruitage innovante, permettant de retrouver un signal perturbé par un bruit à variance non-constante. Cet algorithme est appliqué en finance à la modélisation de la volatilité. Un second travail s'intéresse au dé-bruitage d'un signal perturbé par un bruit asymétrique et leptokurtique. En effet, nous adaptons un modèle de Maximum A Posteriori, couramment employé en imagerie, à des bruits suivant des lois de probabilité de Student, Gaussienne asymétrique et Student asymétrique. Cet algorithme est appliqué au dé-bruitage de prix d'actions haute-fréquences. L'objectif étant d'appliquer un algorithme de reconnaissance de formes sur les extrema locaux du signal dé-bruité. / The main objective of this PhD dissertation is to set up new signal extraction techniques with applications in Finance. In our setting, a signal is defined in two ways. In the framework of investement strategies, a signal is a function which generates buy/sell orders. In denoising theory, a signal, is a function disrupted by some noise, that we want to recover. A first part of this PhD studies historical volatility spillovers around corporate earning announcements. Notably, we study whether a move by one point in the announcer historical volatility in time t will generate a move by beta percents in time t+1. We find evidences of volatility spillovers and we study their intensity across variables such as : the announcement outcome, the surprise effect, the announcer capitalization, the market sentiment regarding the announcer, and other variables. We illustrate our finding by a volatility arbitrage strategy. The second part of the dissertation adapts denoising techniques coming from imagery : wavelets and total variation methods, to forms of noise observed in finance. A first paper proposes an denoising algorithm for a signal disrupted by a noise with a spatially varying standard-deviation. A financial application to volatility modelling is proposed. A second paper adapts the Bayesian representation of the Rudin, Osher and Fatemi approach to asymmetric and leptokurtic noises. A financial application is proposed to the denoising of intra-day stock prices in order to implement a pattern recognition trading strategy.