The Experts below are selected from a list of 511260 Experts worldwide ranked by ideXlab platform
Howard Weiss - One of the best experts on this subject based on the ideXlab platform.
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How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism
Communications in Mathematical Physics, 2008Co-Authors: Mark Pollicott, Howard WeissAbstract:A popular Wavelet reference [W] states that “in theoretical and practical studies, the notion of (Wavelet) regularity has been increasing in importance.” Not surprisingly, the study of Wavelet regularity is currently a major topic of investigation. Smoother Wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct Wavelets converge faster for smoother Wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of Wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported Wavelets, and also applies to non-analytic Wavelet filters. We present an algorithm for computing the Sobolev regularity of Wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of Wavelets that are smoother than the Daubechies Wavelets and have the same support. We establish smooth dependence of the regularity for Wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of Wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional Wavelets.
Mark Pollicott - One of the best experts on this subject based on the ideXlab platform.
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How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism
Communications in Mathematical Physics, 2008Co-Authors: Mark Pollicott, Howard WeissAbstract:A popular Wavelet reference [W] states that “in theoretical and practical studies, the notion of (Wavelet) regularity has been increasing in importance.” Not surprisingly, the study of Wavelet regularity is currently a major topic of investigation. Smoother Wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct Wavelets converge faster for smoother Wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of Wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported Wavelets, and also applies to non-analytic Wavelet filters. We present an algorithm for computing the Sobolev regularity of Wavelets and prove that it converges with super-exponential speed. As an application we construct new examples of Wavelets that are smoother than the Daubechies Wavelets and have the same support. We establish smooth dependence of the regularity for Wavelet families, and we derive a variational formula for the regularity. We also show a general relation between the asymptotic regularity of Wavelet families and maximal measures for the doubling map. Finally, we describe how these results generalize to higher dimensional Wavelets.
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How Smooth is Your Wavelet? Wavelet Regularity via Thermodynamic Formalism
Communications in Mathematical Physics, 2008Co-Authors: Mark Pollicott, Howard M. WeissAbstract:A popular Wavelet reference [W] states that “in theoretical and practical studies, the notion of (Wavelet) regularity has been increasing in importance.” Not surprisingly, the study of Wavelet regularity is currently a major topic of investigation. Smoother Wavelets provide sharper frequency resolution of functions. Also, the iterative algorithms to construct Wavelets converge faster for smoother Wavelets. The main goals of this paper are to extend, refine, and unify the thermodynamic approach to the regularity of Wavelets and to devise a faster algorithm for estimating regularity. The thermodynamic approach works equally well for compactly supported and non-compactly supported Wavelets, and also applies to non-analytic Wavelet filters.
Williams John - One of the best experts on this subject based on the ideXlab platform.
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Wavelet-Galerkin solution of boundary value problems
Archives of Computational Methods in Engineering, 2015Co-Authors: Amaratunga Kevin, Williams JohnAbstract:In this paper we review the application of Wavelets to the solution of partial differential equations. We consider in detail both the single scale and the multiscale Wavelet Galerkin method. The theory of Wavelets is described here using the language and mathematics of signal processing. We show a method of adapting Wavelets to an interval using an extrapolation technique called Wavelet Extrapolation. Wavelets on an interval allow boundary conditions to be enforced in partial differential equations and image boundary problems to be overcome in image processing. Finally, we discuss the fast inversion of matrices arising from differential operators by preconditioning the multiscale Wavelet matrix. Wavelet preconditioning is shown to limit the growth of the matrix condition number, such that Krylov subspace iteration methods can accomplish fast inversion of large matrices.
Valery A. Zheludev - One of the best experts on this subject based on the ideXlab platform.
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periodic orthogonal Wavelets and Wavelet packets
2019Co-Authors: Amir Averbuch, Pekka Neittaanmäki, Valery A. ZheludevAbstract:In this chapter, we discuss how to derive versatile families of periodic discrete-time orthogonal Wavelets and Wavelet packets from discrete and discrete-time splines outlined in Chap. 3. These Wavelets and Wavelet packets, although not having compact supports, are well localized in the time domain. They can have any number of discrete vanishing moments. Their DFT spectra tend to have a rectangular shape when the spline order grows and provide a collection of refined splits of the Nyquist frequency band. The Wavelet and Wavelet packet transforms are implemented in a fast way using the FFT.
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Polynomial Spline-Wavelets
Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, 2015Co-Authors: Amir Averbuch, Pekka Neittaanmäki, Valery A. ZheludevAbstract:This chapter presents Wavelets in the spaces of polynomial splines. The Wavelets’ design is based on the Zak transform, which provides an integral representation of spline-Wavelets. The exponential Wavelets which participate in the integral representation are counterparts of the exponential splines that were introduced in Chap. 4. Fast algorithms for the Wavelet transforms of splines are presented. Generators of spline-Wavelet spaces are described, such as the B-Wavelets and their duals and the Battle-Lemarie Wavelets whose shifts form orthonormal bases of the spline-Wavelet spaces.
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Periodic Spline Wavelets and Wavelet Packets
Spline and Spline Wavelet Methods with Applications to Signal and Image Processing, 2014Co-Authors: Amir Averbuch, Pekka Neittaanmäki, Valery A. ZheludevAbstract:This chapter presents Wavelets and Wavelet packets in the spaces of periodic splines of arbitrary order, which, in essence, are the multiple generators for these spaces. The SHA technique provides explicit representation of the Wavelets and Wavelet packets and fast implementation of the transforms in one and several dimensions.
Amaratunga Kevin - One of the best experts on this subject based on the ideXlab platform.
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Wavelet-Galerkin solution of boundary value problems
Archives of Computational Methods in Engineering, 2015Co-Authors: Amaratunga Kevin, Williams JohnAbstract:In this paper we review the application of Wavelets to the solution of partial differential equations. We consider in detail both the single scale and the multiscale Wavelet Galerkin method. The theory of Wavelets is described here using the language and mathematics of signal processing. We show a method of adapting Wavelets to an interval using an extrapolation technique called Wavelet Extrapolation. Wavelets on an interval allow boundary conditions to be enforced in partial differential equations and image boundary problems to be overcome in image processing. Finally, we discuss the fast inversion of matrices arising from differential operators by preconditioning the multiscale Wavelet matrix. Wavelet preconditioning is shown to limit the growth of the matrix condition number, such that Krylov subspace iteration methods can accomplish fast inversion of large matrices.
