Os sistemas de recuperação de imagens por conteúdo (CBIR -Content-based Image Retrieval) possuem a habilidade de retornar imagens utilizando como chave de busca outras imagens. Considerando uma imagem de consulta, o foco de um sistema CBIR é pesquisar no banco de dados as "n" imagens mais similares à imagem de consulta de acordo com um critério dado. Este trabalho de pesquisa foi direcionado na geração de vetores de características para um sistema CBIR considerando bancos de imagens médicas, para propiciar tal tipo de consulta. Um vetor de características é uma representação numérica sucinta de uma imagem ou parte dela, descrevendo seus detalhes mais representativos. O vetor de características é um vetor "n"-dimensional contendo esses valores. Essa nova representação da imagem pode ser armazenada em uma base de dados, e assim, agilizar o processo de recuperação de imagens. Uma abordagem alternativa para caracterizar imagens para um sistema CBIR é a transformação do domínio. A principal vantagem de uma transformação é sua efetiva caracterização das propriedades locais da imagem. Recentemente, pesquisadores das áreas de matemática aplicada e de processamento de sinais desenvolveram técnicas práticas de "wavelet" para a representação multiescala e análise de sinais. Estas novas ferramentas diferenciam-se das tradicionais técnicas de Fourier pela forma de localizar a informação no plano tempo-freqüência; basicamente...

We are investigating the combination of wavelets and decision trees to detect ships and other maritime surveillance targets from medium resolution SAR images. Wavelets have inherent advantages to extract image descriptors while decision trees are able to handle different data sources. In addition, our work aims to consider oceanic features such as ship wakes and ocean spills. In this incipient work, Haar and Cohen-Daubechies-Feauveau 9/7 wavelets obtain detailed descriptors from targets and ocean features and are inserted with other statistical parameters and wavelets into an oblique decision tree. © 2011 Springer-Verlag.

For the first time, complete source distributions for the emission and absorption of acoustic and electromagnetic wavelets are defined and computed, both in spacetime and Fourier space. The biggest surprise is the great simplicity of the Fourier sources as compared to the rather convoluted spacetime expressions obtained from the original wavelets. This suggests that the associated pulsed-beam propagators may play a fundamental role in emission and absorption processes including focus or "directivity." It also opens the way to constructing FFT-based algorithms for pulsed-beam analyses of acoustic and electromagnetic waves.; Comment: 56 pages, 3 figures. Invited "Topical Review" article for Journal of Physics A: Mathematical and General, http://www.iop.org/journals/jphysa

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. This article gives an overview over some well known results about the continuous and discrete wavelet transforms and groups acting on $\mathbb{R}^n$. We also show how this action can give rise to wavelets, and in particular, MSF wavelets)in $L^2(\mathbb{R}^n)$.

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases.

We study Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent to a multiplication operator with a certain function. Since this function is responsible for many interesting features of these Toeplitz-type operators and their algebras, we investigate its behavior in more detail. As a by-product we obtain an interesting observation about the asymptotic behavior of true polyanalytic Bergman spaces. Isomorphisms between the Calder\'on-Toeplitz operator algebras and functional algebras are described and their consequences are discussed.; Comment: 31 pages, v2: some parts substantially rewritten, 3 metapost figures added, some corrections made, references updated

In this paper a countable family of new compactly supported {\em non-Haar} $p$-adic wavelet bases in ${\cL}^2(\bQ_p^n)$ is constructed. We use the wavelet bases in the following applications: in the theory of $p$-adic pseudo-differential operators and equations. Namely, we study the connections between wavelet analysis and spectral analysis of $p$-adic pseudo-differential operators. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. In addition, $p$-adic wavelets are used to construct solutions of linear and semi-linear pseudo-differential equations. Since many $p$-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models.

The purpose of this paper is to articulate an observation that many interesting type of wavelets (or coherent states) arise from group representations which are not square integrable or vacuum vectors which are not admissible. This extends an applicability of the popular wavelets construction to classic examples like the Hardy space. Keywords: Wavelets, coherent states, group representations, Hardy space, functional calculus, Berezin calculus, Radon transform, Moebius map, maximal function, affine group, special linear group, numerical range.; Comment: 7 pages, LaTeX2e

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,d\nu_\lambda) and L^2_h(C^4_+,d\tilde\nu_\lambda) of square integrable holomorphic functions with scale dimension \lambda and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a \lambda-extension of Schwinger's Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L^2_h(D_4...

New continuous wavelets of compact support are introduced, which are related to the beta distribution. They can be built from probability distributions using 'blur'derivatives. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters a and b. Close expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem applied for compactly supported signals.; Comment: 7 pages, 4 figures. Journal of Communication and Information Systems, former Journal of the Brazilian Telecommunications Society, ISSN 1980-6604 pp.105-111

A new family of wavelets is introduced, which is associated with Legendre polynomials. These wavelets, termed spherical harmonic or Legendre wavelets, possess compact support. The method for the wavelet construction is derived from the association of ordinary second order differential equations with multiresolution filters. The low-pass filter associated with Legendre multiresolution analysis is a linear phase finite impulse response filter (FIR).; Comment: 6 pages, 6 figures, 1 table In: Computational Methods in Circuits and Systems Applications, WSEAS press, pp.211-215, 2003. ISBN: 960-8052-88-2

We introduce a method to construct large classes of MSF wavelets of the Hardy space H^2(\R) and symmetric MSF wavelets of L^2(\R), and discuss the classification of such sets. As application, we show that there are uncountably many wavelet sets of L^2(\R) and H^2(\R). We also enumerate all symmetric wavelets of L^2(\R) with at most three intervals in the positive axis as well as 3-interval wavelet sets of H^2(\R). Finally, we construct families of MSF wavelets of L^2(\R) whose Fourier transform does not vanish in any neighbourhood of the origin.; Comment: 28 pages

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.; Comment: 63 pages with 22 ps-figures...

Here we present a method of constructing steerable wavelet frames in $L_2(\mathbb{R}^d)$ that generalizes and unifies previous approaches, including Simoncelli's pyramid and Riesz wavelets. The motivation for steerable wavelets is the need to more accurately account for the orientation of data. Such wavelets can be constructed by decomposing an isotropic mother wavelet into a finite collection of oriented mother wavelets. The key to this construction is that the angular decomposition is an isometry, whereby the new collection of wavelets maintains the frame bounds of the original one. The general method that we propose here is based on partitions of unity involving spherical harmonics. A fundamental aspect of this construction is that Fourier multipliers composed of spherical harmonics correspond to singular integrals in the spatial domain. Such transforms have been studied extensively in the field of harmonic analysis, and we take advantage of this wealth of knowledge to make the proposed construction practically feasible and computationally efficient.; Comment: 27 pages, 2 figures

We present an application of error theory using Dirichlet Forms in linear partial differential equations (LPDE). We study the transmission of an uncertainty on the terminal condition to the solution of the LPDE thanks to the decomposition of the solution on a wavelets basis. We analyze the basic properties and a particular class of LPDE where the wavelets bases show their powerful, the combination of error theory and wavelets basis justifies some hypotheses, helpful to simplify the computation.; Comment: 17 pages, some misprints corrected

We provide a characterization of wavelets on local fields of positive characteristic based on results on affine and quasi affine frames. This result generalizes the characterization of wavelets on Euclidean spaces by means of two basic equations. We also give another characterization of wavelets. Further, all wavelets which are associated with a multiresolution analysis on a such a local field are also characterized.; Comment: arXiv admin note: text overlap with arXiv:1103.0090

Some connections between operator theory and wavelet analysis: Since the mid eighties, it has become clear that key tools in wavelet analysis rely crucially on operator theory. While isolated variations of wavelets, and wavelet constructions had previously been known, since Haar in 1910, it was the advent of multiresolutions, and subband filtering techniques which provided the tools for our ability to now easily create efficient algorithms, ready for a rich variety of applications to practical tasks. Part of the underpinning for this development in wavelet analysis is operator theory. This will be presented in the lectures, and we will also point to a number of developments in operator theory which in turn derive from wavelet problems, but which are of independent interest in mathematics. Some of the material will build on chapters in a new wavelet book, co-authored by the speaker and Ola Bratteli, see http://www.math.uiowa.edu/~jorgen/ .; Comment: 63 pages, 10 figures/tables, LaTeX2e ("mrv9x6" document class), Contribution by Palle E. T. Jorgensen to the Tutorial Sessions, Program: ``Functional and harmonic analyses of wavelets and frames,'' 4-7 August 2004, Organizers: Judith Packer, Qiyu Sun, Wai Shing Tang. v2 adds Section 2.3.4...

Using the wavelet theory introduced by the author and J. Benedetto, we present examples of wavelets on p-adic fields and other locally compact abelian groups with compact open subgroups. We observe that in this setting, the Haar and Shannon wavelets (which are at opposite extremes over the real numbers) coincide and are localized both in time and in frequency. We also study the behavior of the translation operators required in the theory.; Comment: 21 pages; LaTeX2e; to appear in the proceedings of the AMS Special Session on Wavelets, Frames, and Operator Theory held at Baltimore, January 2003

A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\bR^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct Latin square wavelets as rank 2 variants of Haar wavelets. Also we construct nonseparable scaling functions for rank 2 variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.; Comment: 35 pages, 3 figures

This survey paper examines the work of J. von Neumann and M.H. Stone as it relates to the abstract theory of wavelets. In particular, we discuss the direct integral theory of von Neumann and how it can be applied to representations of certain discrete groups to study the existence of normalized tight frames in the setting of Gabor systems and wavelets, via the use of group representations and von Neumann algebras. Then the extension of Stone's theorem due to M. Naimark, W. Ambrose and R. Godement is reviewed, and its relationship to the multiresolution analyses of S. Mallat and Y. Meyer and the generalized multiresolution analyses of L. Baggett, H. Medina, and K. Merrill. Finally, the paper ends by discussing some recent work due to the author, Baggett, P. Jorgensen and Merrill, and its relationship to operator theory.; Comment: 27 pages, will appear in Contemporary Mathematics Volume edited by R. Doran and R. Kadison, minor corrections added