Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES); Pós-graduação em Matematica Aplicada e Computacional - FCT; Nesta dissertação é realizado um estudo comparativo sobre alguns tipos de algoritmos aplicados a imagens digitais voltado para interpolação. Este trabalho inclui os métodos clássicos, que são: replicação, bilinear, bicúbica, Lagrange e interpolação pela função sinc; e alguns recentes: algoritmo-localmente adaptativo, método New Edge-Direction Interpolation (NEDI), improved New Edge Direction Interpolation (iNEDI), iterative curvaturebased interpolation (ICBI), interpolação utilizando wavelets redundantes e utilizando filtro bilateral. Todos os novos métodos possuem melhorias em aspectos visuais e redução de ruídos nas bordas em relação aos clássicos. Os métodos avaliados são comparados visualmente e quantitativamente utilizando as métricas estatísticas: erro médio quadrático (MSE), Raíz do Erro Médio Quadrático (RMSE), Erro Médio Quadrático Normalizado (NMSE), Relação Sinal-Ruído (SNR), Coeficiente de Correlação (CC) e Índice de Qualidade Universal (IQI). Também é realizada uma discussão dos resultados obtidos, analisando as qualidades e os defeitos dos métodos estudados. Por fim...

Sin lugar a dudas los métodos wavelets permiten desarrollar algoritmos eficientes y novedosos en el estudio del procesamiento de imágenes y señales. La idea de utilizar esta teoría en la solución numérica de ecuaciones en derivadas parciales se da en virtud a que algunas propiedades de las wavelets son importantes en la construcción de algoritmos adaptativos. Un algoritmo de este tipo selecciona un conjunto minimal de aproximaciones en cada paso, de tal manera que la solución calculada sea lo suficientemente próxima a la solución exacta. Si queremos que la solución calculada sea suave en alguna región, sólo unos pocos coeficientes wavelet serán necesarios para obtener una buena aproximación de la solución en dicha región, es decir, solamente los coeficientes de bajas frecuencias cuyo soporte esté en esa región son los utilizados. De otro lado, los coeficientes grandes (en valor absoluto) se localizan cerca de las singularidades y esto nos permite definir criterios de adaptabilidad a través del tiempo de evaluación [15, 23, 53, 64]. Este trabajo se dirige fundamentalmente a encontrar soluciones aproximadas a problemas del tipo hiperbólico o parabólicos, utilizando el método wavelet-Galerkin. El trabajo busca dar respuesta problemas que surgen en diferentes áreas de las ciencias e ingeniería.; v...

In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras O_N, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L^2(T) by (S_i\xi)(z)=m_i(z)\xi(z^N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over L^2(T). This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations of O_N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.; Comment: 59 pages, AMS-LaTeX v1.2b

Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in $L^2(\mathbb{R}^d)$ which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over $\mathbb{R}^d$, $d >1$. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Han's proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in $\hat{\mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.; Comment: 15 pages

The paper develops theory of covariant transform, which is inspired by the wavelet construction. It was observed that many interesting types of wavelets (or coherent states) arise from group representations which are not square integrable or vacuum vectors which are not admissible. Covariant transform extends an applicability of the popular wavelets construction to classic examples like the Hardy space H_2, Banach spaces, covariant functional calculus and many others. Keywords: Wavelets, coherent states, group representations, Hardy space, Littlewood-Paley operator, functional calculus, Berezin calculus, Radon transform, Moebius map, maximal function, affine group, special linear group, numerical range, characteristic function, functional model.; Comment: 9 pages, LaTeX2e (AMS-LaTeX); v2: minor corrections

This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The ``wavelet transform'' maps each $f(x)$ to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higher-order wavelets are constructed, and it is surprisingly quick to compute with them --- always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including high-definition television). So far the Fourier Transform --- or its 8 by 8 windowed version, the Discrete Cosine Transform --- is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory.; Comment: 18 pages

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.; Comment: 34 pages

A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis.

We use Lorentz polynomials to present the solutions explicitly of equations (6.1.7) of [I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992] and (4.9) of [I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988), no. 7, 909--996] sot that we give an efficient way to prove Daubechies' results on the existence of spline type orthogonal scaling functions and to evaluate Daubechies scaling functions.

Particle physics has for some time made extensive use of extended field configuations such as solitons, instantons, and sphalerons. However, no direct use has yet been made of the quite extensive literature on ``localized wave'' configurations developed by the engineering, optics, and mathematics communities. In this article I will exhibit a particularly simple ``physical wavelet'' -- it is a Lorentz covariant classical field configuration that lives in physical Minkowski space. The field is everwhere finite and nonsingular, and has quadratic falloff in both space and time. The total energy is finite, the total action is zero, and the field configuration solves the wave equation. These physical wavelets can be constructed for both complex and real scalar fields, and can be extended to the Maxwell and Yang-Mills fields in a straightforward manner. Since these wavelets are finite energy, they are guaranteed to be classically present at finite temperature; since they are zero action, they can contribute to the quantum mechanical path integral at zero ``cost''.; Comment: 12 pages, JHEP3.cls

Recently, compressed sensing techniques in combination with both wavelet and directional representation systems have been very effectively applied to the problem of image inpainting. However, a mathematical analysis of these techniques which reveals the underlying geometrical content is completely missing. In this paper, we provide the first comprehensive analysis in the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. First, we propose an abstract model for problems of data recovery and derive error bounds for two different recovery schemes, namely l_1 minimization and thresholding. Second, we set up a particular microlocal model for an image governed by edges inspired by seismic data as well as a particular mask to model the missing data, namely a linear singularity masked by a horizontal strip. Applying the abstract estimate in the case of wavelets and of shearlets we prove that -- provided the size of the missing part is asymptotically to the size of the analyzing functions -- asymptotically precise inpainting can be obtained for this model. Finally, we show that shearlets can fill strictly larger gaps than wavelets in this model.; Comment: 49 pages...

This paper offers a new regard on compactly supported wavelets derived from FIR filters. Although being continuous wavelets, analytical formulation are lacking for such wavelets. Close approximations for daublets (Daubechies wavelets) and their spectra are introduced here. The frequency detection properties of daublets are investigated through scalograms derived from these new analytical expressions. These near-daublets have been implemented on the Matlab wavelet toolbox and a few scalograms presented. This approach can be valuable for wavelet synthesis from hardware or for application involving continuous wavelet-based systems, such as wavelet OFDM.; Comment: 6 pages, 6 figures, 3 tables. Conference: International Telecommunication Symposium, ITS 2010, Manaus, AM , Brazil

The first objective of this paper is to introduce a unified approach to the D/A conversion, a real-time algorithm referred to as {\it blending operator}, based on spline functions of arbitrarily desired order, to interpolate the irregular data samples, while preserving all polynomials of the same spline order, with assured maximum order of approximation. This helps remove the two main obstacles for adapting the recently proposed time-frequency analysis technique {\it Synchrosqueezing transform} (SST) to irregular data samples in order to allow online computation. Secondly, for real-time dynamic information extraction from an oscillatory signal via SST, a family of vanishing-moment and minimum-supported spline-wavelets (to be called VM wavelets) are introduced for on-line computation of the CWT and its derivative. The second objective of this paper is to apply the proposed real-time algorithm and VM wavelets to clinical applications, particularly to the study of the "anesthetic depth" of a patient during surgery, with emphasis on analyzing two dynamic quantities: the "instantaneous frequencies" and the "non-rhythmic to rhythmic ratios" of the patient's respiration, based on a one-lead electrocardiogram (ECG) signal.It is envisioned that the proposed algorithm and VM wavelets should enable real-time monitoring of "anesthetic depth"...

We study a class of localized solutions of the wave equation, called eigenwavelets, obtained by extending its fundamental solutions to complex spacetime in the sense of hyperfunctions. The imaginary spacetime variables y, which form a timelike vector, act as scale parameters generalizing the scale variable of wavelets in one dimension. They determine the shape of the wavelets in spacetime, making them pulsed beams that can be focused as tightly as desired around a single ray by letting y approach the light cone. Furthermore, the absence of any sidelobes makes them especially attractive for communications, remote sensing and other applications using acoustic waves. (A similar set of "electromagnetic eigenwavelets" exists for Maxwell's equations.) I review the basic ideas in Minkowski space, then compute sources whose realization should make it possible to radiate and absorb such wavelets. This motivates an extension of Huygens' principle allowing equivalent sources to be represented on shells instead of surfaces surrounding a bounded source.; Comment: 15 pages, 4 figures, invited paper, conference honoring Carlos Berenstein

Let $\mathscr Q$ be the quaternion Heisenberg group, and let $\mathbf P$ be the affine automorphism group of $\mathscr Q$. We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of $\mathbf P$ on $L^2(\mathscr Q)$. A class of radial wavelets is constructed. The inverse wavelet transform is simplified by using radial wavelets. Then we investigate the Radon transform on $\mathscr Q$. A Semyanistri-Lizorkin space is introduced, on which the Radon transform is a bijection. We deal with the Radon transform on $\mathscr Q$ both by the Euclidean Fourier transform and the group Fourier transform. These two treatments are essentially equivalent. We also give an inversion formula by using wavelets, which does not require the smoothness of functions if the wavelet is smooth.

Recent works emphasized the interest of numerical solution of PDE's with wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non linear approximation aspect of the wavelet approximation of PDE's to prove the relevance of such methods. In order to extend these results, we focuss on the convergence of the iterative algorithm, and we consider different possibilities offered by the wavelet theory: the tensorial wavelets and the derivation/integration of wavelet bases. We also investigate the use of wavelet packets. We apply these extended results to prove in the case of the Shannon wavelets, the convergence of the Leray projector algorithm with divergence-free wavelets.; Comment: preprint IMPAN (19 pages)

The quantum mechanical harmonic oscillator Hamiltonian generates a one-parameter unitary group W(\theta) in L^2(R) which rotates the time-frequency plane. In particular, W(\pi/2) is the Fourier transform. When W(\theta) is applied to any frame of Gabor wavelets, the result is another such frame with identical frame bounds. Thus each Gabor frame gives rise to a one-parameter family of frames, which we call a deformation of the original. For example, beginning with the usual tight frame F of Gabor wavelets generated by a compactly supported window g(t) and parameterized by a regular lattice in the time-frequency plane, one obtains a family of frames F_\theta generated by the non-compactly supported windows g_\theta=W(theta)g, parameterized by rotated versions of the original lattice. This gives a method for constructing tight frames of Gabor wavelets for which neither the window nor its Fourier transform have compact support. When \theta=\pi/2, we obtain the well-known Gabor frame generated by a window with compactly supported Fourier transform. The family F_\theta therefore interpolates these two familiar examples.; Comment: 8 pages in Plain Tex

In this paper novel classes of 2-D vector-valued spatial domain wavelets are defined, and their properties given. The wavelets are 2-D generalizations of 1-D analytic wavelets, developed from the Generalized Cauchy-Riemann equations and represented as quaternionic functions. Higher dimensionality complicates the issue of analyticity, more than one `analytic' extension of a real function is possible, and an `analytic' analysis wavelet will not necessarily construct `analytic' decomposition coefficients. The decomposition of locally unidirectional and/or separable variation is investigated in detail, and two distinct families of hyperanalytic wavelet coefficients are introduced, the monogenic and the hypercomplex wavelet coefficients. The recasting of the analysis in a different frame of reference and its effect on the constructed coefficients is investigated, important issues for sampled transform coefficients. The magnitudes of the coefficients are shown to exhibit stability with respect to shifts in phase. Hyperanalytic 2-D wavelet coefficients enable the retrieval of a phase-and-magnitude description of an image in phase space, similarly to the description of a 1-D signal with the use of 1-D analytic wavelets, especially appropriate for oscillatory signals. Existing 2-D directional wavelet decompositions are related to the newly developed framework...

Let $({\mathcal X},d,\mu)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hyt\"onen, together with obtaining some crucial lower bounds for regular wavelets, the authors give an unconditional basis of $H^1_{\rm at}({\mathcal X})$ and several equivalent characterizations of $H^1_{\rm at}({\mathcal X})$ in terms of wavelets, which are proved useful.; Comment: 40 pages, submitted. We spilit the article arXiv:1506.05910 into two papers and this is the first one

We construct a Continuous Wavelet Transform (CWT) on the torus $\mathbb T^2$ following a group-theoretical approach based on the conformal group $SO(2,2)$. The Euclidean limit reproduces wavelets on the plane $\mathbb R^2$ with two dilations, which can be defined through the natural tensor product representation of usual wavelets on $\mathbb R$. Restricting ourselves to a single dilation imposes severe conditions for the mother wavelet that can be overcome by adding extra modular group $SL(2,\mathbb Z)$ transformations, thus leading to the concept of \emph{modular wavelets}. We define modular-admissible functions and prove frame conditions.; Comment: 21 pages, 10 figures