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## ‣ Multidimensional smoothing using hyperbolic interpolatory wavelets

Hegland, M; Nielsen, Ole; Shen, Zuowei
Tipo: Artigo de Revista Científica Formato: 13 pages
Português
Relevância na Pesquisa
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Affiliation in article: Nielsen, Ole, Urban Risk Research Group, Geoscience Australia, Canberra

## ‣ Projective multiresolution analyses for $L^2(R^2)$

Packer, Judith A.; Rieffel, Marc A.
Tipo: Artigo de Revista Científica
Português
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We define the notion of "projective" multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra $C(\btn)$ of continuous complex-valued functions on an $n$-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for $L^2(\brn)$. Then we show how to construct examples for the case of any diagonal $2 \times 2$ dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective $C(\mathbb T^2)$-module. We compute the isomorphism classes of the corresponding wavelet modules.; Comment: 25 pages

## ‣ A connection between multiresolution wavelet theory of scale N and representations of the Cuntz algebra O_N

Bratteli, Ola; Jorgensen, Palle E. T.
Tipo: Artigo de Revista Científica
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In this paper we give a short survey of a connection between the theory of wavelets in L^2(R) and certain representations of the Cuntz algebra on L^2(T).; Comment: 13 pages, AMS-TeX version 2.1, uses LaTeX circle font lcircle10. To appear in J. Roberts, ed., Proceedings of the Rome Conference on Operator Algebras and Quantum Field Theory. Survey article; for complete proofs see funct-an/9612002 and funct-an/9612003 by the same authors

## ‣ Pure states on O_d

Bratteli, Ola; Jorgensen, Palle E. T.; Kishimoto, Akitaka; Werner, Reinhard F.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We study representations of the Cuntz algebras O_d and their associated decompositions. In the case that these representations are irreducible, their restrictions to the gauge-invariant subalgebra UHF_d have an interesting cyclic structure. If S_i, 1 \leq i \leq d, are representatives of the Cuntz relations on a Hilbert space H, special attention is given to the subspaces which are invariant under S_i^*. The applications include wavelet multiresolutions corresponding to wavelets of compact support (to appear in the later paper \cite{BEJ97}), and finitely correlated states on one-dimensional quantum spin chains.; Comment: 42 pages, 0 figures, AMS-LaTeX v1.2b

## ‣ Deconvolution of confocal microscopy images using proximal iteration and sparse representations

Dupé, François-Xavier; Fadili, Jalal; Starck, Jean Luc
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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We propose a deconvolution algorithm for images blurred and degraded by a Poisson noise. The algorithm uses a fast proximal backward-forward splitting iteration. This iteration minimizes an energy which combines a \textit{non-linear} data fidelity term, adapted to Poisson noise, and a non-smooth sparsity-promoting regularization (e.g $\ell_1$-norm) over the image representation coefficients in some dictionary of transforms (e.g. wavelets, curvelets). Our results on simulated microscopy images of neurons and cells are confronted to some state-of-the-art algorithms. They show that our approach is very competitive, and as expected, the importance of the non-linearity due to Poisson noise is more salient at low and medium intensities. Finally an experiment on real fluorescent confocal microscopy data is reported.

## ‣ A multi-level preconditioned Krylov method for the efficient solution of algebraic tomographic reconstruction problems

Cools, Siegfried; Ghysels, Pieter; van Aarle, Wim; Sijbers, J.; Vanroose, Wim
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction.; Comment: Journal of Computational and Applied Mathematics (2014)...

## ‣ Detection of objects in noisy images and site percolation on square lattices

Langovoy, Mikhail A.; Wittich, Olaf
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We propose a novel probabilistic method for detection of objects in noisy images. The method uses results from percolation and random graph theories. We present an algorithm that allows to detect objects of unknown shapes in the presence of random noise. Our procedure substantially differs from wavelets-based algorithms. The algorithm has linear complexity and exponential accuracy and is appropriate for real-time systems. We prove results on consistency and algorithmic complexity of our procedure.; Comment: This paper first appeared as EURANDOM Report 2009-035 on November 11, 2009. Link to the paper at the EURANDOM repository: http://www.eurandom.tue.nl/reports/2009/035-report.pdf Link to the abstract at EURANDOM repository: http://www.eurandom.tue.nl/reports/2009/035-abstract.pdf

## ‣ Detection of objects in noisy images based on percolation theory

Davies, Patrick Laurie; Langovoy, Mikhail A.; Wittich, Olaf
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We propose a novel statistical method for detection of objects in noisy images. The method uses results from percolation and random graph theories. We present an algorithm that allows to detect objects of unknown shapes in the presence of nonparametric noise of unknown level. The noise density is assumed to be unknown and can be very irregular. Our procedure substantially differs from wavelets-based algorithms. The algorithm has linear complexity and exponential accuracy and is appropriate for real-time systems. We prove results on consistency and algorithmic complexity of our procedure.; Comment: Submitted for publication on January 19, 2010

## ‣ Fuzzy sets in nonparametric Bayes regression

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A simple Bayesian approach to nonparametric regression is described using fuzzy sets and membership functions. Membership functions are interpreted as likelihood functions for the unknown regression function, so that with the help of a reference prior they can be transformed to prior density functions. The unknown regression function is decomposed into wavelets and a hierarchical Bayesian approach is employed for making inferences on the resulting wavelet coefficients.; Comment: Published in at http://dx.doi.org/10.1214/074921708000000084 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

## ‣ Association schemes on general measure spaces and zero-dimensional Abelian groups

Barg, Alexander; Skriganov, Maxim
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. In this paper we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator realizations of the adjacency algebras of schemes and derive simple properties of these algebras. To develop a theory of general association schemes, we focus on schemes on topological Abelian groups where we can employ duality theory and the machinery of harmonic analysis. We construct translation association schemes on such groups using the language of spectrally dual partitions. Such partitions are shown to arise naturally on topological zero-dimensional Abelian groups, for instance, Cantor-type groups or the groups of p-adic numbers. This enables us to construct large classes of dual pairs of association schemes on zero-dimensional groups with respect to their Haar measure, and to compute their eigenvalues and intersection numbers. We also derive properties of infinite metric schemes, connecting them with the properties of the non-Archimedean metric on the group. Pursuing the connection between schemes on zero-dimensional groups and harmonic analysis, we show that the eigenvalues have a natural interpretation in terms of Littlewood-Paley wavelet bases...

## ‣ On boundedness of Calder\'on-Toeplitz operators

Hutník, Ondrej
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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We study the boundedness of Toeplitz-type operators defined in the context of the Calder\'on reproducing formula considering the specific wavelets whose Fourier transforms are related to Laguerre polynomials. Some sufficient conditions for simultaneous boundedness of these Calder\'on-Toeplitz operators on each wavelet subspace for unbounded symbols are given, where investigating the behavior of certain sequence of iterated integrals of symbols is helpful. A number of examples and counterexamples is given.; Comment: 17 pages, v2: small changes after the referee's report and typos corrected

## ‣ Hausdorff dimension of boundaries of self-affine tiles in R^n

Veerman, J. J. P.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.888838%
We present a new method to calculate the Hausdorff dimension of a certain class of fractals: boundaries of self-affine tiles. Among the interesting aspects are that even if the affine contraction underlying the iterated function system is not conjugated to a similarity we obtain an upper- and lower-bounds for its Hausdorff dimension. In fact, we obtain the exact value for the dimension if the moduli of the eigenvalues of the underlying affine contraction are all equal (this includes Jordan blocks). The tiles we discuss play an important role in the theory of wavelets. We calculate the dimension for a number of examples.

## ‣ Noncommutative analysis, Multivariable spectral theory for operators in Hilbert space, Probability, and Unitary Representations

Jorgensen, Palle; Tian, Feng
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.888838%
Over the decades, Functional Analysis has been enriched and inspired on account of demands from neighboring fields, within mathematics, harmonic analysis (wavelets and signal processing), numerical analysis (finite element methods, discretization), PDEs (diffusion equations, scattering theory), representation theory; iterated function systems (fractals, Julia sets, chaotic dynamical systems), ergodic theory, operator algebras, and many more. And neighboring areas, probability/statistics (for example stochastic processes, Ito and Malliavin calculus), physics (representation of Lie groups, quantum field theory), and spectral theory for Schr\"odinger operators. We have strived for a more accessible book, and yet aimed squarely at applications; -- we have been serious about motivation: Rather than beginning with the four big theorems in Functional Analysis, our point of departure is an initial choice of topics from applications. And we have aimed for flexibility of use; acknowledging that students and instructors will invariably have a host of diverse goals in teaching beginning analysis courses. And students come to the course with a varied background. Indeed, over the years we found that students have come to the Functional Analysis sequence from other and different areas of math...

## ‣ Antieigenvalue Analysis, New Applications: Continuum Mechanics, Economics, Number Theory

Gustafson, Karl
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.888838%
My recent book Antieigenvalue Analysis, World-Scientific, 2012, presented the theory of antieigenvalues from its inception in 1966 up to 2010, and its applications within those forty-five years to Numerical Analysis, Wavelets, Statistics, Quantum Mechanics, Finance, and Optimization. Here I am able to offer three further areas of application: Continuum Mechanics, Economics, and Number Theory.

## ‣ Definability and stability of multiscale decompositions for manifold-valued data

Grohs, Philipp; Wallner, Johannes
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midpoint-interpolating. For definable multiscale decompositions we obtain a stability result.

## ‣ Multiresolution in the Bergman space

Pap, Margit
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.888838%
In this paper we give a multiresolution construction in Bergman space. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may cause a drawback of the method. We give a set of poles and using them we will generate a multiresolution in $A^2$. We study the upper and lower density of this set, and we give sufficient conditions for this set to be interpolating or sampling sequence for the Bergman space. The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the Bergman space. The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to the Bergman space if we can measure their values on a given set of points inside the unit disc. Convergence properties of the hyperbolic wavelet representation will be studied.; Comment: 15 pages, revised version

## ‣ Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections

Giné, Evarist; Nickl, Richard
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.888838%
Given an i.i.d. sample from a distribution $F$ on $\mathbb{R}$ with uniformly continuous density $p_0$, purely data-driven estimators are constructed that efficiently estimate $F$ in sup-norm loss and simultaneously estimate $p_0$ at the best possible rate of convergence over H\"older balls, also in sup-norm loss. The estimators are obtained by applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-type analogs of the inequalities in Koltchinskii [Ann. Statist. 34 (2006) 2593-2656] for the deviation of suprema of empirical processes from their Rademacher symmetrizations.; Comment: Published in at http://dx.doi.org/10.3150/09-BEJ239 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

## ‣ Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition

Postnikov, Eugene B.; Lebedeva, Elena A.; Lavrova, Anastasia I.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Recently, it has been proven [R. Soc. Open Sci. 1 (2014) 140124] that the continuous wavelet transform with non-admissible kernels (approximate wavelets) allows for an existence of the exact inverse transform. Here we consider the computational possibility for the realization of this approach. We provide modified simpler explanation of the reconstruction formula, restricted on the practical case of real valued finite (or periodic/periodized) samples and the standard (restricted) Morlet wavelet as a practically important example of an approximate wavelet. The provided examples of applications includes the test function and the non-stationary electro-physical signals arising in the problem of neuroscience.; Comment: 18 pages, 4 figures

## ‣ Multiresolution analysis for Markov Interval Maps

Bohnstengel, Jana; Kesseböhmer, Marc
Tipo: Artigo de Revista Científica
We set up a multiresolution analysis on fractal sets derived from limit sets of Markov Interval Maps. For this we consider the $\mathbb{Z}$-convolution of a non-atomic measure supported on the limit set of such systems and give a thorough investigation of the space of square integrable functions with respect to this measure. We define an abstract multiresolution analysis, prove the existence of mother wavelets, and then apply these abstract results to Markov Interval Maps. Even though, in our setting the corresponding scaling operators are in general not unitary we are able to give a complete description of the multiresolution analysis in terms of multiwavelets.; Comment: 31 pages, 4 figures