Página 11 dos resultados de 435 itens digitais encontrados em 0.010 segundos

## ‣ Uso de ôndulas na compressão de operadores; Using wavelets for operator compression

Nunes, Ana Luísa da Silva
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## ‣ Position Coding

Aboufadel, Edward; Armstrong, Timothy; Smietana, Elizabeth
Tipo: Artigo de Revista Científica
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A position coding pattern is an array of symbols in which subarrays of a certain fixed size appear at most once. So, each subarray uniquely identifies a location in the larger array, which means there is a bijection of some sort from this set of subarrays to a set of coordinates. The key to Fly Pentop Computer paper and other examples of position codes is a method to read the subarray and then convert it to coordinates. Position coding makes use of ideas from discrete mathematics and number theory. In this paper, we will describe the underlying mathematics of two position codes, one being the Anoto code that is the basis of "Fly paper". Then, we will present two new codes, one which uses binary wavelets as part of the bijection.; Comment: 14 pages, 7 figures

## ‣ Positive definite maps, representations and frames

Dutkay, Dorin Ervin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We present a unitary approach to the construction of representations and intertwining operators. We apply it to the $C^*$-algebras, groups, Gabor type unitary systems and wavelets. We give an application of our method to the theory of frames, and we prove a general dilation theorem which is in turn applied to specific cases, and we obtain in this way a dilation theorem for wavelets.

## ‣ Ergodic optimization of prevalent super-continuous functions

Bochi, Jairo; Zhang, Yiwei
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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Given a dynamical system, we say that a performance function has property P if its time averages along orbits are maximized at a periodic orbit. It is conjectured by several authors that for sufficiently hyperbolic dynamical systems, property P should be typical among sufficiently regular performance functions. In this paper we address this problem using a probabilistic notion of typicality that is suitable to infinite dimension: the concept of prevalence as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two symbols, we prove that property P is prevalent in spaces of functions with a strong modulus of regularity. Our proof uses Haar wavelets to approximate the ergodic optimization problem by a finite-dimensional one, which can be conveniently restated as a maximum cycle mean problem on a de Bruijin graph.; Comment: 23 pages, 4 figures. Final version, to appear in International Mathematics Research Notices (IMRN)

## ‣ Multiwavelet density estimation

Locke, Judson B.; Peter, Adrian M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.15793%
Accurate density estimation methodologies play an integral role in a variety of scientific disciplines, with applications including simulation models, decision support tools, and exploratory data analysis. In the past, histograms and kernel density estimators have been the predominant tools of choice, primarily due to their ease of use and mathematical simplicity. More recently, the use of wavelets for density estimation has gained in popularity due to their ability to approximate a large class of functions, including those with localized, abrupt variations. However, a well-known attribute of wavelet bases is that they can not be simultaneously symmetric, orthogonal, and compactly supported. Multiwavelets-a more general, vector-valued, construction of wavelets-overcome this disadvantage, making them natural choices for estimating density functions, many of which exhibit local symmetries around features such as a mode. We extend the methodology of wavelet density estimation to use multiwavelet bases and illustrate several empirical results where multiwavelet estimators outperform their wavelet counterparts at coarser resolution levels.; Comment: Preprint

## ‣ Taylor Series as Wide-sense Biorthogonal Wavelet Decomposition

de Oliveira, H. M.; Lins, R. D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Pointwise-supported generalized wavelets are introduced, based on Dirac, doublet and further derivatives of delta. A generalized biorthogonal analysis leads to standard Taylor series and new Dual-Taylor series that may be interpreted as Laurent Schwartz distributions. A Parseval-like identity is also derived for Taylor series, showing that Taylor series support an energy theorem. New representations for signals called derivagrams are introduced, which are similar to spectrograms. This approach corroborates the impact of wavelets in modern signal analysis.; Comment: 6 pages, 4 figures. conference: XXII Simposio Brasileiro de Telecomunicacoes, SBrT'05, 2005, Campinas, SP, Brazil

## ‣ Phase retrieval for the Cauchy wavelet transform

Mallat, Stéphane; Waldspurger, Irène
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.15793%
We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the unicity and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in $L^2$-norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact. Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question.

## ‣ Some recent advances on the RBF

Chen, W
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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This paper aims to survey our recent work relating to the radial basis function (RBF) and its applications to numerical PDEs. We introduced the kernel RBF involving general pre-wavelets and scale-orthogonal wavelets RBF. A dimension-independent RBF error bound was also conjectured. The centrosymmetric structure of RBF interpolation matrix under symmetric sample knots was pointed out. On the other hand, we introduced the boundary knot method via nonsingular general solution and dual reciprocity principle and the boundary particle method via multiple reciprocity principle. By using the Green integral we developed a domain-type Hermite RBF scheme called the modified Kansa method, which significantly reduces calculation errors around boundary. To circumvent the Gibbs phenomenon, the least square RBF collocation scheme was presented. All above discretization schemes are meshfree, symmetric, spectral convergent, integration-free and mathematically very simple. The numerical validations are also briefly presented.

## ‣ Orthonormal dilations of non-tight frames

Bownik, Marcin; Jasper, John; Speegle, Darrin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We establish dilation theorems for non-tight frames with additional structure, i.e., frames generated by unitary groups of operators and projective unitary representations. This generalizes previous dilation results for Parseval frames due to Han and Larson and Gabardo and Han. We also extend the dilation theorem for Parseval wavelets, due to Dutkay, Han, Picioroaga, and Sun , by identifying the optimal class of frame wavelets for which dilation into an orthonormal wavelet is possible.

## ‣ Stable and robust sampling strategies for compressive imaging

Krahmer, Felix; Ward, Rachel
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.15793%
In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence -- the so-called local coherence -- measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by $\ell_1$-minimization and by total variation minimization. The local coherence framework developed in this paper should be of independent interest in sparse recovery problems more generally...

## ‣ Square Roots of -1 in Real Clifford Algebras

Hitzer, Eckhard; Helmstetter, Jacques; Ablamowicz, Rafal
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [32] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\mathbb{R}^3$. Further research on general algebras $Cl(p,q)$ has explicitly derived the geometric roots of -1 for $p+q \leq 4$ [17]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of -1 found in the different types of Clifford algebras, depending on the type of associated ring ($\mathbb{R}$, $\mathbb{H}$, $\mathbb{R}^2$, $\mathbb{H}^2$, or $\mathbb{C}$). At the end of the paper explicit computer generated tables of representative square roots of -1 are given for all Clifford algebras with $n=5,7$, and $s=3 \, (mod 4)$ with the associated ring $\mathbb{C}$. This includes, e.g., $Cl(0,5)$ important in Clifford analysis, and $Cl(4,1)$ which in applications is at the foundation of conformal geometric algebra. All these roots of -1 are immediately useful in the construction of new types of geometric Clifford Fourier transformations.; Comment: 31 pages...

## ‣ Unitary systems and wavelet sets

Larson, David R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We give a detailed description of the local commutant approach to wavelet theory using operator algebraic methods. We include a new result on interpolation pairs of wavelet sets: Every pair in the generalized Journe family of wavelet sets is an interpolation pair. In the last section, we give a discussion of several open problems on wavelets and frame-wavelets.; Comment: This is a semi-expository article which was written by the author for the proceedings of the 4th International Conference on Wavelet Analysis and its Applications, Macau, China, December 2005 [WAA2005], as a keynote speaker; 30p

## ‣ Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements

Adcock, Ben; Hansen, Anders C.; Kutyniok, Gitta; Ma, Jackie
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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In this paper we analyze two-dimensional wavelet reconstructions from Fourier samples within the framework of generalized sampling. For this, we consider both separable compactly-supported wavelets and boundary wavelets. We prove that the number of samples that must be acquired to ensure a stable and accurate reconstruction scales linearly with the number of reconstructing wavelet functions. We also provide numerical experiments that corroborate our theoretical results.

## ‣ A hypergeometric basis for the Alpert multiresolution analysis

Geronimo, Jeffrey S.; Iliev, Plamen
Tipo: Artigo de Revista Científica
Português
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We construct an explicit orthonormal basis of piecewise ${}_{i+1}F_{i}$ hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of ${}_2F_3$ hypergeometric functions. Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced ${}_4 F_3$ hypergeometric functions evaluated at $1$, which allows to compute them recursively via three-term recurrence relations. The above results lead to a variety of new interesting identities and orthogonality relations reminiscent to classical identities of higher-order hypergeometric functions and orthogonality relations of Wigner $6j$-symbols.

## ‣ The local trace function of shift invariant subspaces

Dutkay, Dorin Ervin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.15793%
We define the local trace function for subspaces of $\ltworn$ which are invariant under integer translation. Our trace function contains the dimension function and the spectral function defined by Bownik and Rzeszotnik and completely characterizes the given translation invariant subspace. It has properties such as positivity, additivity, monotony and some form of continuity. It behaves nicely under dilations and modulations. We use the local trace function to deduce, using short and simple arguments, some fundamental facts about wavelets such as the characterizing equations, the equality between the dimension function and the multiplicity function and some new relations between scaling functions and wavelets.

## ‣ Prime Coset Sum: A Systematic Method for Designing Multi-D Wavelet Filter Banks with Fast Algorithms

Hur, Youngmi; Zheng, Fang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.15793%
As constructing multi-D wavelets remains a challenging problem, we propose a new method called prime coset sum to construct multi-D wavelets. Our method provides a systematic way to construct multi-D non-separable wavelet filter banks from two 1-D lowpass filters, with one of whom being interpolatory. Our method has many important features including the following: 1) it works for any spatial dimension, and any prime scalar dilation, 2) the vanishing moments of the multi-D wavelet filter banks are guaranteed by certain properties of the initial 1-D lowpass filters, and furthermore, 3) the resulting multi-D wavelet filter banks are associated with fast algorithms that are faster than the existing fast tensor product algorithms.

## ‣ Reproducing subgroups of Sp(2,R). Part II: admissible vectors

Alberti, Giovanni S.; De Mari, Filippo; De Vito, Ernesto; Mantovani, Lucia
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of $Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here, we determine for which of these groups the restriction of the metaplectic representation gives rise to a reproducing formula. In all the positive cases we characterize the admissible vectors with a generalized Calder\'on equation. They include products of 1D-wavelets, directional wavelets, shearlets, and many new examples.; Comment: 37 pages, 3 figures

## ‣ On generic frequency decomposition. Part 1: vectorial decomposition

Vergara, Sossio
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The famous Fourier theorem states that, under some restrictions, any periodic function (or real world signal) can be obtained as a sum of sinusoids, and hence, a technique exists for decomposing a signal into its sinusoidal components. From this theory an entire branch of research has flourished: from the Short-Time or Windowed Fourier Transform to the Wavelets, the Frames, and lately the Generic Frequency Analysis. The aim of this paper is to take the Frequency Analysis a step further. It will be shown that keeping the same reconstruction algorithm as the Fourier Theorem but changing to a new computing method for the analysis phase allows the generalization of the Fourier Theorem to a large class of nonorthogonal bases. New methods and algorithms can be employed in function decomposition on such generic bases. It will be shown that these algorithms are a generalization of the Fourier analysis, i.e. they are reduced to the familiar Fourier tools when using orthogonal bases. The differences between this tool and the wavelets and frames theories will be discussed. Examples of analysis and reconstruction of functions using the given algorithms and nonorthogonal bases will be given. In this first part the focus will be on vectorial decomposition...

## ‣ Asymptotic Analysis of Inpainting via Universal Shearlet Systems

Genzel, Martin; Kutyniok, Gitta
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.15793%
Recently introduced inpainting algorithms using a combination of applied harmonic analysis and compressed sensing have turned out to be very successful. One key ingredient is a carefully chosen representation system which provides (optimally) sparse approximations of the original image. Due to the common assumption that images are typically governed by anisotropic features, directional representation systems have often been utilized. One prominent example of this class are shearlets, which have the additional benefitallowing faithful implementations. Numerical results show that shearlets significantly outperform wavelets in inpainting tasks. One of those software packages, www.shearlab.org, even offers the flexibility of usingdifferent parameter for each scale, which is not yet covered by shearlet theory. In this paper, we first introduce universal shearlet systems which are associated with an arbitrary scaling sequence, thereby modeling the previously mentioned flexibility. In addition, this novel construction allows for a smooth transition between wavelets and shearlets and therefore enables us to analyze them in a uniform fashion. For a large class of such scaling sequences, we first prove that the associated universal shearlet systems form band-limited Parseval frames for $L^2(\mathbb{R}^2)$ consisting of Schwartz functions. Secondly...

## ‣ Wavelet filter functions, the matrix completion problem, and projective modules over $C(\mathbb T^n)$

Packer, Judith A.; Rieffel, Marc A.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.985708%
We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective $C(\mathbb T^n)$-modules. Conversely, we show how cancellation properties for finitely generated projective modules over $C(\mathbb T^n)$ can often be used to prove the existence of continuous high pass filters, of the kind needed for multivariate wavelets, corresponding to a given continuous low-pass filter. However, we also give an example of a continuous low-pass filter for which it is impossible to find corresponding continuous high-pass filters. In this way we give another approach to the solution of the matrix completion problem for filters of the kind arising in wavelet theory.; Comment: 21 pages, various local improvements