This work presents a novel approach in order to increase the recognition power of Multiscale Fractal Dimension (MFD) techniques, when applied to image classification. The proposal uses Functional Data Analysis (FDA) with the aim of enhancing the MFD technique precision achieving a more representative descriptors vector, capable of recognizing and characterizing more precisely objects in an image. FDA is applied to signatures extracted by using the Bouligand-Minkowsky MFD technique in the generation of a descriptors vector from them. For the evaluation of the obtained improvement, an experiment using two datasets of objects was carried out. A dataset was used of characters shapes (26 characters of the Latin alphabet) carrying different levels of controlled noise and a dataset of fish images contours. A comparison with the use of the well-known methods of Fourier and wavelets descriptors was performed with the aim of verifying the performance of FDA method. The descriptor vectors were submitted to Linear Discriminant Analysis (LDA) classification method and we compared the correctness rate in the classification process among the descriptors methods. The results demonstrate that FDA overcomes the literature methods (Fourier and wavelets) in the processing of information extracted from the MFD signature. In this way...

This article is dedicated to harmonic wavelet Galerkin methods for the solution of partial differential equations. Several variants of the method are proposed and analyzed, using the Burgers equation as a test model. The computational complexity can be reduced when the localization properties of the wavelets and restricted interactions between different scales are exploited. The resulting variants of the method have computational complexities ranging from O(N(3)) to O(N) (N being the space dimension) per time step. A pseudo-spectral wavelet scheme is also described and compared to the methods based on connection coefficients. The harmonic wavelet Galerkin scheme is applied to a nonlinear model for the propagation of precipitation fronts, with the front locations being exposed in the sizes of the localized wavelet coefficients. (C) 2011 Elsevier Ltd. All rights reserved.; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq

Uma característica importante do tráfego gerado pelo protocolo Internet Protocol (IP) é a existência de padrões scaling, que impactam significantemente o desempenho dos mecanismos de controle de tráfego e que, por isto vem sendo foco de atenção de diversas pesquisas. A natureza scaling do tráfego IP tem sido alvo de bastante polêmica. Em pequenas escalas de tempo o tráfego IP é altamente variável e a variabilidade difere da natureza fractal encontrada em grandes escalas de tempo, não existindo ainda um consenso em relação a natureza do tráfego IP nestas pequenas escalas de tempo. No presente estudo, foram revisadas as evidências da multifractalidade nas pequenas escalas de tempo, através da análise experimental de diversos traços de tráfego real. Constatou-se que não se pode generalizar a natureza dos fluxos do protocolo IP e do protocolo Transmission Control Protocol (TCP) como monofractal ou como multifractal, enquanto que a natureza do fluxo do protocolo User Datagram Protocol (UDP) é sempre multifractal. O crescente uso do protocolo UDP pelas emergentes aplicações que necessitam requisitos de tempo real altera consideravelmente a natureza scaling do tráfego IP, dado que este tipo de tráfego n?ao reage a situações de congestionamento. Apesar de existirem diversos modelos para tráfego TCP...

Este trabalho apresenta um procedimento de detecção do fenômeno de rubbing e seus efeitos em máquinas rotativas, aplicável para o regime estacionário e não-estacionário, utilizando como ferramenta de análise a Transformada de Wavelet. Primeiramente, aplica-se a Transformada de Wavelet Contínua (CWT) para sinais estacionários com o uso da wavelet Morlet Complexa como wavelet mãe, para extração de características intrínsecas da falha estudada. Aplica-se esta técnica para este regime de operação, pois, a análise multi-resolução é capaz de evidenciar possíveis transientes, mesmo no regime estacionário. Posteriormente, aplicou-se a Transformada de Wavelet Packet (WPT) visando-se analisar a variação da distribuição das energias contidas em bandas de freqüências específicas para o regime nãoestacionário. Diante da grande quantidade de informações contidas no sinal de partida e passagem pelas velocidades críticas do sistema, esta segunda técnica foi escolhida considerandose sua habilidade de compactação de dados, viabilizando o uso de outras técnicas e também o monitoramento on-line de máquinas. Todas as análises em questão, CWT no regime estacionário e a WPT no não-estacionário, são através de simulações computacionais com auxílio do Método dos Elementos Finitos e para sinais reais proveniente de uma bancada experimental de testes; This work presents a detection procedure of the rubbing phenomenon and its effects in rotating machinery...

Os substratos que constituem um reservatório são identificados com a utilização de dados sísmicos. O processo consiste na determinação de propriedades petrofísicas dos reservatórios. Entretanto, camadas de estruturas subsísmica são mascaradas devido à limitação do dado sísmico convencional. A falha proporcionada por esses dados produz um modelo comprometido para o reservatório. A falta de resolução vertical do dado sísmico pode ser superada através da integração do mesmo com todas as informações disponíveis do reservatório (Sancevero, 2006). A maneira mais efetiva de reunir esses dados encontra-se na geração de modelos de impedância acústica originados do processo de inversão. Como foco principal, foi produzido um algoritmo de inversão sísmica para impedância acústica, manuseado com o software MATLAB. Nesse algoritmo, o dado sísmico foi modelado como sendo a convolução da resposta refletiva do meio com o pulso sísmico sonoro, conhecido como wavelet. A assinatura do pulso sonoro faz com que o dado sísmico seja de banda limitada no domínio da frequência. O método proposto permite acrescentar nos dados sísmicos valores de baixas frequências providos de dados de perfis de poços. Tal método também permite analisar o impacto da extração da wavelet por meio da amarração sísmica-poço...

O resumo poderá ser visualizado no texto completo da tese digital.; The abstract is available with the full electronic document.

This work presents a method for automatic classification of faults and events related to quality of service in power distribution networks, based on oscillographies of the bar feeder voltages of the distribution substation. We present the results for two distinct pre-processing forms of the voltage signals. The first is based on the Fourier Transform and the second on the Wavelet Transform for different families of wavelet functions. We compared three neural models for the process of classification: Multi-Layer Perceptron, Radial Basis Function and Support Vector Machine. The models were trained taking into account the autonomous operation of networks, i.e. automatic model selection and control complexity. The results were validated for a set of simulations performed using the Alternative Transient Program, aimed at practical implementation of the proposed method in an oscillograph logger, developed by Lactec together with Copel, called the Power Quality Monitor. The results were obtained with performance on the order of 90% of average accuracy for the various pre-processing forms and neural models.; Este trabalho apresenta uma metodologia para classificação de eventos de curto-circuito e mano-bras em redes de distribuição de energia elétrica...

Throughout this article the major idea and conclusion is about comparing this method with some very famous methods like fourier series and wavelet, to show that the power of this approximation method is as much as to predicate many natural and finance methods, something which we can not say the same for wavelets and Fourier series, since this method consider the function itself to make the base functions, and it is more natural rather than wavelets method and fourier series, which they consider some prior functions as basis.; Comment: 14 pages, 11 fig, 3 table

We give an operator theoretic approach to the constructions of multiresolutions as they are used in a number of basis constructions with wavelets, and in Hilbert spaces on fractals. Our approach starts with the following version of the classical Baumslag-Solitar relations $u t = t^2 u$ where $t$ is a unitary operator in a Hilbert space $\mathcal H$ and $u$ is an isometry in $\mathcal H$. There are isometric dilations of this system into a bigger Hilbert space, relevant for wavelets. For a variety of carefully selected dilations, the ``bigger'' Hilbert space may be $L^2(\br)$, and the dilated operators may be the unitary operators which define a dyadic wavelet multiresolutions of $L^2(\br)$ with the subspace $\mathcal H$ serving as the corresponding resolution subspace. That is, the initialized resolution which is generated by the wavelet scaling function(s). In the dilated Hilbert space, the Baumslag-Solitar relations then take the more familiar form $u t u^{-1} = t^2$. We offer an operator theoretic framework including the standard construction; and we show how the representations of certain discrete semidirect group products serve to classify the possibilities. For this we analyze and compare several types of unitary representations of these semidirect products: the induced representations in Mackey's theory...

Several years ago, O. Bratelli and P. Jorgensen developed the concept of m-systems of filters for dilation by a positive integer N>1 on L^2(R). They constructed a loop group action on m-systems. By work of Mallat and Meyer, these m-systems are important in constructing multi-resolution analyses and wavelets associated to dilation by N and translation by Z on L^2(R). In this paper, we discuss an extension of this loop-group construction to generalized filter systems, which we will call ``M-systems,'' associated with generalized multiresolution analyses. In particular, we show that every multiplicity function has an associated generalized loop group which acts freely and transitively on the set of M-systems corresponding to the multiplicity function. The results of Bratteli and Jorgensen correspond to the case where the multiplicity function is identically equal to 1.; Comment: 15 pages; AMS-LaTeX; submitted to proceedings of AMS Special Session on Wavelets, Frames, and Operator Theory held at Baltimore

In this paper, we characterize the symbol in H\"ormander symbol class $S^{m}_{\rho,\delta} (m\in R, \rho,\delta\geq 0)$ by its wavelet coefficients. Consequently, we analyse the kernel-distribution property for the symbol in the symbol class $S^{m}_{\rho,\delta} (m\in R, \rho>0, \delta\geq 0)$ which is more general than known results; for non-regular symbol operators, we establish sharp $L^{2}$-continuity which is better than Calder\'on and Vaillancourt's result, and establish $L^{p} (1\leq p\leq\infty)$ continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator's continuity on the basis of the wavelets coefficients in phase space.; Comment: 22 pages

Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers of a matrix. In this paper we consider the following simultaneous tiling problem: Given a lattice in $\L\in \R^d$ and a matrix $A\in\GLd$, does there exist a measurable set $T$ such that both $\{T+\alpha: \alpha\in\L\}$ and $\{A^nT: n\in\Z\}$ are tilings of $\R^d$? This problem comes directly from the study of wavelets and wavelet sets. Such a $T$ is known to exist if $A$ is expanding. When $A$ is not expanding the problem becomes much more subtle. Speegle \cite{Spe03} exhibited examples in which such a $T$ exists for some $\L$ and nonexpanding $A$ in $\R^2$. In this paper we give a complete solution to this problem in $\R^2$.; Comment: 16 pages, no figures

We study orthogonality relations for Fourier frequencies and complex exponentials in Hilbert spaces $L^2(\mu)$ with measures $\mu$ arising from iterated function systems (IFS). This includes equilibrium measures in complex dynamics. Motivated by applications, we draw parallels between analysis of fractal measures on the one hand, and the geometry of wavelets on the other. We are motivated by spectral theory for commuting partial differential operators and related duality notions. While stated initially for bounded and open regions in $\br^d$, they have since found reformulations in the theory of fractals and wavelets. We include a historical sketch with questions from early operator theory.; Comment: v2, minor correction in section 4

We show that a class of dynamical systems induces an associated operator system in Hilbert space. The dynamical systems are defined from a fixed finite-to-one mapping in a compact metric space, and the induced operators form a covariant system in a Hilbert space of L^2-martingales. Our martingale construction depends on a prescribed set of transition probabilities, given by a non-negative function. Our main theorem describes the induced martingale systems completely. The applications of our theorem include wavelets, the dynamics defined by iterations of rational functions, and sub-shifts in symbolic dynamics. In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. While our motivation derives from some wavelet problems, we have in mind other applications as well; and the issues involving covariant operator systems may be of independent interest.; Comment: 44 pages, LaTeX2e ("jotart" document class); v2: A few opening paragraphs were added to the paper; an addition where a bit of the history is explained...

Orthogonal wavelets, or wavelet frames, for L^2(R) are associated with quadrature mirror filters (QMF). The latter constitute a set of complex numbers which relate the dyadic scaling of functions on R to the Z-translates, and which satisfy the QMF-axioms. In this paper, we show that generically, the data in the QMF-systems of wavelets is minimal, in the sense that it cannot be nontrivially reduced. The minimality property is given a geometric formulation in the Hilbert space l^2(Z), and it is then shown that minimality corresponds to irreducibility of a wavelet representation of the algebra O_2; and so our result is that this family of representations of O_2 on the Hilbert space l^2(Z) is irreducible for a generic set of values of the parameters which label the wavelet representations.; Comment: LaTeX2e amsart class; 69 pages, 2 tables, 4 figures, 12 pages of plots (total 162 EPS graphics); full-resolution EPS graphics available at ftp://ftp.math.uiowa.edu/pub/jorgen/MinimalityWavelet Changes: correction in Theorem 5.9 and its proof; some added implications and clarifications; slightly more condensed, with an expanded introduction and more graphics in a smaller format. Accepted for publication in Advances in Mathematics

Wavelets and their associated transforms are highly efficient when approximating and analyzing one-dimensional signals. However, multivariate signals such as images or videos typically exhibit curvilinear singularities, which wavelets are provably deficient of sparsely approximating and also of analyzing in the sense of, for instance, detecting their direction. Shearlets are a directional representation system extending the wavelet framework, which overcomes those deficiencies. Similar to wavelets, shearlets allow a faithful implementation and fast associated transforms. In this paper, we will introduce a comprehensive carefully documented software package coined ShearLab 3D (www.ShearLab.org) and discuss its algorithmic details. This package provides MATLAB code for a novel faithful algorithmic realization of the 2D and 3D shearlet transform (and their inverses) associated with compactly supported universal shearlet systems incorporating the option of using CUDA. We will present extensive numerical experiments in 2D and 3D concerning denoising, inpainting, and feature extraction, comparing the performance of ShearLab 3D with similar transform-based algorithms such as curvelets, contourlets, or surfacelets. In the spirit of reproducible reseaerch...

Classical multiscale analysis based on wavelets has a number of successful applications, e.g. in data compression, fast algorithms, and noise removal. Wavelets, however, are adapted to point singularities, and many phenomena in several variables exhibit intermediate-dimensional singularities, such as edges, filaments, and sheets. This suggests that in higher dimensions, wavelets ought to be replaced in certain applications by multiscale analysis adapted to intermediate-dimensional singularities. My lecture described various initial attempts in this direction. In particular, I discussed two approaches to geometric multiscale analysis originally arising in the work of Harmonic Analysts Hart Smith and Peter Jones (and others): (a) a directional wavelet transform based on parabolic dilations; and (b) analysis via anistropic strips. Perhaps surprisingly, these tools have potential applications in data compression, inverse problems, noise removal, and signal detection; applied mathematicians, statisticians, and engineers are eagerly pursuing these leads.

Within the area of applied harmonic analysis, various multiscale systems such as wavelets, ridgelets, curvelets, and shearlets have been introduced and successfully applied. The key property of each of those systems are their (optimal) approximation properties in terms of the decay of the $L^2$-error of the best $N$-term approximation for a certain class of functions. In this paper, we introduce the general framework of $\alpha$-molecules, which encompasses most multiscale systems from applied harmonic analysis, in particular, wavelets, ridgelets, curvelets, and shearlets as well as extensions of such with $\alpha$ being a parameter measuring the degree of anisotropy, as a means to allow a unified treatment of approximation results within this area. Based on an $\alpha$-scaled index distance, we first prove that two systems of $\alpha$-molecules are almost orthogonal. This leads to a general methodology to transfer approximation results within this framework, provided that certain consistency and time-frequency localization conditions of the involved systems of $\alpha$-molecules are satisfied. We finally utilize these results to enable the derivation of optimal sparse approximation results \msch{for} a specific class of cartoon-like functions by sufficient conditions on the 'control' parameters of a system of $\alpha$-molecules.

In this paper it is shown the performing of an optical transform to state the scalar diffraction in the formulation of the wavelet transform and the 'wave equations'. From there, a bridge is build between equations of spherical waves presented in 1678 by Huygens and the continuous wavelet transform. For such a purpose, wavelets are introduced that meet the principles of waves and the properties of wavelets. The following equations are applied in solution to show a correspondence between the Huygens-Fresnel diffraction and the wavelet transform.; Comment: 5 pages, 3 figures

We show how for $n=2,3 (\mod 4)$ continuous Clifford (geometric) algebra (GA) $Cl_n$-valued admissible wavelets can be constructed using the similitude group $SIM(n)$. We strictly aim for real geometric interpretation, and replace the imaginary unit $i \in \C$ therefore with a GA blade squaring to $-1$. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a $Cl_{n}$ Clifford Fourier Transform and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform. As an example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle.; Comment: 4 pages