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## ‣ Denosing Using Wavelets and Projections onto the L1-Ball

Cetin, A. Enis; Tofighi, Mohammad
Tipo: Artigo de Revista Científica
Publicado em 10/06/2014 Português
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Both wavelet denoising and denosing methods using the concept of sparsity are based on soft-thresholding. In sparsity based denoising methods, it is assumed that the original signal is sparse in some transform domains such as the wavelet domain and the wavelet subsignals of the noisy signal are projected onto L1-balls to reduce noise. In this lecture note, it is shown that the size of the L1-ball or equivalently the soft threshold value can be determined using linear algebra. The key step is an orthogonal projection onto the epigraph set of the L1-norm cost function.; Comment: Submitted to Signal Processing Magazine

## ‣ Bivariate Daubechies Scaling Functions (Wavelets)

Aboufadel, Edward; Cox, Amanda; Zee, Amy Vander
Tipo: Artigo de Revista Científica
Publicado em 28/03/2001 Português
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Using the Daubechies conditions of compact support, orthogonal, and regularity, we were able to derive bivariate scaling functions with which to reproduce linear functions (planes). We describe how to create all possible masks of refinement coefficients that satisfy those conditions.; Comment: 10 pages

## ‣ Geophysical modelling with Colombeau functions: Microlocal properties and Zygmund regularity

Hoermann, G.; de Hoop, M. V.
Tipo: Artigo de Revista Científica
Português
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In global seismology Earth's properties of fractal nature occur. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality. For the purpose of seismic wave propagation, we model the Earth's properties as Colombeau generalized functions. In one spatial dimension, we have a precise characterization of Zygmund regularity in Colombeau algebras. This is made possible via a relation between mollifiers and wavelets.; Comment: corrected Definition 13; updated Remark 5

## ‣ Linear independence of compactly supported separable shearlet systems

Ma, Jackie; Petersen, Philipp
Tipo: Artigo de Revista Científica
Português
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This paper examines linear independence of shearlet systems. This property has already been studied for wavelets and other systems such as, for instance, for Gabor systems. In fact, for Gabor systems this problem is commonly known as the HRT conjecture. In this paper we present a proof of linear independence of compactly supported separable shearlet systems. For this, we employ a sampling strategy to utilize the structure of an implicitly given underlying oversampled wavelet system as well as the shape of the supports of the shearlet elements.

## ‣ Sparse Principal Components Analysis

Johnstone, Iain M; Lu, Arthur Yu
Tipo: Artigo de Revista Científica
Publicado em 27/01/2009 Português
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Principal components analysis (PCA) is a classical method for the reduction of dimensionality of data in the form of n observations (or cases) of a vector with p variables. For a simple model of factor analysis type, it is proved that ordinary PCA can produce a consistent (for n large) estimate of the principal factor if and only if p(n) is asymptotically of smaller order than n. There may be a basis in which typical signals have sparse representations: most co-ordinates have small signal energies. If such a basis (e.g. wavelets) is used to represent the signals, then the variation in many coordinates is likely to be small. Consequently, we study a simple "sparse PCA" algorithm: select a subset of coordinates of largest variance, estimate eigenvectors from PCA on the selected subset, threshold and reexpress in the original basis. We illustrate the algorithm on some exercise ECG data, and prove that in a single factor model, under an appropriate sparsity assumption, it yields consistent estimates of the principal factor.; Comment: This manuscript was written in late 2003; a much revised version is to appear, with discussion and later references, in the Journal of the American Statistical Association in 2009. The JASA revision cites this archive version for the proof of inconsistency of PCA

## ‣ On the Translation Invariance of Wavelet Subspaces

Weber, Eric
Tipo: Artigo de Revista Científica
Publicado em 22/09/1999 Português
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An examination of the translation invariance of $V_0$ under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.; Comment: 11 pages, AMS-Latex

## ‣ Construction of bivariate symmetric orthonormal wavelets with short support

Lai, Ming-Jun; Roach, David W.
Tipo: Artigo de Revista Científica
Publicado em 29/09/2011 Português
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In this paper, we give a parameterization of the class of bivariate symmetric orthonormal scaling functions with filter size $6\times 6$ using the standard dilation matrix 2I. In addition, we give two families of refinable functions which are not orthonormal but have associated tight frames. Finally, we show that the class of bivariate symmetric scaling functions with filter size $8\times 8$ can not have two or more vanishing moments.; Comment: 33 pages

## ‣ Separable representations, KMS states, and wavelets for higher-rank graphs

Farsi, Carla; Gillaspy, Elizabeth; Kang, Sooran; Packer, Judith
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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Let $\Lambda$ be a strongly connected, finite higher-rank graph. In this paper, we construct representations of $C^*(\Lambda)$ on certain separable Hilbert spaces of the form $L^2(X,\mu)$, by introducing the notion of a $\Lambda$-semibranching function system (a generalization of the semibranching function systems studied by Marcolli and Paolucci). In particular, when $\Lambda$ is aperiodic, we obtain a faithful representation of $C^*(\Lambda)$ on $L^2(\Lambda^\infty, M)$, where $M$ is the Perron-Frobenius probability measure on the infinite path space $\Lambda^\infty$ recently studied by an Huef, Laca, Raeburn, and Sims. We also show how a $\Lambda$-semibranching function system gives rise to KMS states for $C^*(\Lambda)$. For the higher-rank graphs of Robertson and Steger, we also obtain a representation of $C^*(\Lambda)$ on $L^2(X, \mu)$, where $X$ is a fractal subspace of $[0,1]$ by embedding $\Lambda^{\infty}$ into $[0,1]$ as a fractal subset $X$ of $[0,1]$. In this latter case we additionally show that there exists a KMS state for $C^*(\Lambda)$ whose inverse temperature is equal to the Hausdorff dimension of $X$. Finally, we construct a wavelet system for $L^2(\Lambda^\infty, M)$ by generalizing the work of Marcolli and Paolucci from graphs to higher-rank graphs.; Comment: Modified hypotheses in Theorem 3.5

## ‣ Needlet algorithms for estimation in inverse problems

Kerkyacharian, Gérard; Petrushev, Pencho; Picard, Dominique; Willer, Thomas
Tipo: Artigo de Revista Científica
Publicado em 02/05/2007 Português
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We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames (termed "needlets") built upon the SVD bases. We consider two different situations: the "wavelet" scenario, where the needlets are assumed to behave similarly to true wavelets, and the "Jacobi-type" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the $L_2$ case, we obtain interesting rates of convergence for other $L_p$ norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEED-D estimator outperforms other standard algorithms in almost all situations.; Comment: Published at http://dx.doi.org/10.1214/07-EJS014 in the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)

## ‣ Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits

Ullrich, Tino
Tipo: Artigo de Revista Científica
Português
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We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions and spaces involving the Peetre maximal function to apply the classical coorbit space theory due to Feichtinger and Gr\"ochenig. This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.

## ‣ A note on reassignment for wavelets

Reimann, Hans Martin
Tipo: Artigo de Revista Científica
Publicado em 29/09/2015 Português
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The reassignment method for the wavelet transform is investigated. Particularly good results are obtained if the wavelet is an extremal for the uncertainty relation of the affine group.

## ‣ Oracle inequalities and minimax rates for non-local means and related adaptive kernel-based methods

Arias-Castro, Ery; Salmon, Joseph; Willett, Rebecca
Tipo: Artigo de Revista Científica
Português
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This paper describes a novel theoretical characterization of the performance of non-local means (NLM) for noise removal. NLM has proven effective in a variety of empirical studies, but little is understood fundamentally about how it performs relative to classical methods based on wavelets or how various parameters (e.g., patch size) should be chosen. For cartoon images and images which may contain thin features and regular textures, the error decay rates of NLM are derived and compared with those of linear filtering, oracle estimators, variable-bandwidth kernel methods, Yaroslavsky's filter and wavelet thresholding estimators. The trade-off between global and local search for matching patches is examined, and the bias reduction associated with the local polynomial regression version of NLM is analyzed. The theoretical results are validated via simulations for 2D images corrupted by additive white Gaussian noise.; Comment: 49 pages, 15 figures

## ‣ Mysteries around the graph Laplacian eigenvalue 4

Nakatsukasa, Yuji; Saito, Naoki; Woei, Ernest
Tipo: Artigo de Revista Científica
Português
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We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue distribution for such a tree is a smooth bell-shaped curve starting from the eigenvalue 0 up to 4. Then, at the eigenvalue 4, there is a sudden jump. Interestingly, the eigenvectors corresponding to the eigenvalues below 4 are semi-global oscillations (like Fourier modes) over the entire tree or one of the branches; on the other hand, those corresponding to the eigenvalues above 4 are much more localized and concentrated (like wavelets) around junctions/branching vertices. For a special class of trees called starlike trees, we obtain a complete understanding of such phase transition phenomenon. For a general graph, we prove the number of the eigenvalues larger than 4 is bounded from above by the number of vertices whose degrees is strictly higher than 2. Moreover, we also prove that if a graph contains a branching path, then the magnitudes of the components of any eigenvector corresponding to the eigenvalue greater than 4 decay exponentially from the branching vertex toward the leaf of that branch.; Comment: 22 pages

## ‣ A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements

Adcock, Ben; Hansen, Anders C.; Roman, Bogdan
Tipo: Artigo de Revista Científica
Português
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This note complements the paper "The quest for optimal sampling: Computationally efficient, structure-exploiting measurements for compressed sensing" [2]. Its purpose is to present a proof of a result stated therein concerning the recovery via compressed sensing of a signal that has structured sparsity in a Haar wavelet basis when sampled using a multilevel-subsampled discrete Fourier transform. In doing so, it provides a simple exposition of the proof in the case of Haar wavelets and discrete Fourier samples of more general result recently provided in the paper "Breaking the coherence barrier: A new theory for compressed sensing" [1].; Comment: 8 pages, companion paper

## ‣ Estimating the joint distribution of independent categorical variables via model selection

Durot, C.; Lebarbier, E.; Tocquet, A. -S.
Tipo: Artigo de Revista Científica
Publicado em 12/06/2009 Português
Relevância na Pesquisa
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Assume one observes independent categorical variables or, equivalently, one observes the corresponding multinomial variables. Estimating the distribution of the observed sequence amounts to estimating the expectation of the multinomial sequence. A new estimator for this mean is proposed that is nonparametric, non-asymptotic and implementable even for large sequences. It is a penalized least-squares estimator based on wavelets, with a penalization term inspired by papers of Birg\'{e} and Massart. The estimator is proved to satisfy an oracle inequality and to be adaptive in the minimax sense over a class of Besov bodies. The method is embedded in a general framework which allows us to recover also an existing method for segmentation. Beyond theoretical results, a simulation study is reported and an application on real data is provided.; Comment: Published in at http://dx.doi.org/10.3150/08-BEJ155 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

## ‣ The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising

Valkonen, Tuomo
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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Let $u \in \mbox{BV}(\Omega)$ solve the total variation denoising problem with $L^2$-squared fidelity and data $f$. Caselles et al. [Multiscale Model. Simul. 6 (2008), 879--894] have shown the containment $\mathcal{H}^{m-1}(J_u \setminus J_f)=0$ of the jump set $J_u$ of $u$ in that of $f$. Their proof unfortunately depends heavily on the co-area formula, as do many results in this area, and as such is not directly extensible to higher-order, curvature-based, and other advanced geometric regularisers, such as total generalised variation (TGV) and Euler's elastica. These have received increased attention in recent times due to their better practical regularisation properties compared to conventional total variation or wavelets. We prove analogous jump set containment properties for a general class of regularisers. We do this with novel Lipschitz transformation techniques, and do not require the co-area formula. In the present Part 1 we demonstrate the general technique on first-order regularisers, while in Part 2 we will extend it to higher-order regularisers. In particular, we concentrate in this part on TV and, as a novelty, Huber-regularised TV. We also demonstrate that the technique would apply to non-convex TV models as well as the Perona-Malik anisotropic diffusion...

## ‣ On construction of multivariate symmetric MRA-based wavelets

Krivoshein, A.
Tipo: Artigo de Revista Científica
Publicado em 12/01/2012 Português
Relevância na Pesquisa
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For an arbitrary matrix dilation, any integer n and any integer/semi-integer c, we describe all masks that are symmetric with respect to the point c and have sum rule of order n. For each such mask, we give explicit formulas for wavelet functions that are point symmetric/antisymmetric and generate frame-like wavelet system providing approximation order n. For any matrix dilations (which are appropriate for axial symmetry group on R^2 in some natural sense) and given integer n, axial symmetric/antisymmetric frame-like wavelet systems providing approximation order n are constructed. Also, for several matrix dilations the explicit construction of highly symmetric frame-like wavelet systems providing approximation order n is presented.

## ‣ A geometric approach to the cascade approximation operator for wavelets

Jorgensen, Palle E. T.