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A real algebra perspective on multivariate tight wavelet frames

TL;DR: In this article, the existence of multivariate tight wavelet frames with at least one vanishing moment was shown to be possible in dimension at most 2 and in dimension ≥ 3, respectively.
Abstract: Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle by Ron and Shen are interpreted in terms of hermitian sums of squares of certain nonnegative trigonometric polynomials and in terms of semi-definite programming. The latter together with the recent results in algebraic geometry and semi-definite programming allow us to answer affirmatively the long standing open question of the existence of such tight wavelet frames in dimension $d=2$; we also provide numerically efficient methods for checking their existence and actual construction in any dimension. We exhibit a class of counterexamples in dimension $d=3$ showing that, in general, the UEP property is not sufficient for the existence of tight wavelet frames. On the other hand we provide stronger sufficient conditions for the existence of tight wavelet frames in dimension $d > 3$ and illustrate our results by several examples.
Citations
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Journal ArticleDOI
TL;DR: A surprise announcement by then prime minister Tony Blair in 1998 led to the creation of a new senior clinical role - the nurse consultant.
Abstract: A surprise announcement by then prime minister Tony Blair in 1998 led to the creation of a new senior clinical role - the nurse consultant.

462 citations

Journal ArticleDOI
TL;DR: In this paper, the authors derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient for an extension with just two generators.
Abstract: The unitary extension principle by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the -type wavelet systems that can be extended to a Parseval wavelet frame by adding just one -type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

15 citations


Cites background from "A real algebra perspective on multi..."

  • ...An interesting discussion of the complexity of the extension problem for wavelet systems in higher dimensions, together with several deep results, recently appeared in [2]....

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Journal ArticleDOI
TL;DR: This work considers the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provides a positive answer under extra assumptions about whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames.
Abstract: It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual (multi-) wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples that demonstrate the extra flexibility in the extension to dual pairs of frames compared with the more popular extensions to tight frames.

9 citations


Cites background from "A real algebra perspective on multi..."

  • ..., the papers [3, 6, 14, 17, 19], just to mention a few out of many)....

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Book ChapterDOI
01 Jan 2014
TL;DR: In this article, a follow-up on the article Frames and Extension Problems I is presented, where the authors go into more recent progress on the topic and also present some open problems.
Abstract: This article is a follow-up on the article Frames and Extension Problems I. Here we will go into more recent progress on the topic and also present some open problems.

2 citations

Posted Content
TL;DR: In this paper, it is shown that a pair of Bessel sequences with wavelet structure can be extended to dual frames by adding a singly generated wavelet system under extra assumptions.
Abstract: It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual multiscale wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples showing that extensions to dual frame pairs are attractive because they often allow better properties than the more popular extensions to tight frames.
References
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Journal ArticleDOI
TL;DR: In this article, the regularity of compactly supported wavelets and symmetry of wavelet bases are discussed. But the authors focus on the orthonormal bases of wavelets, rather than the continuous wavelet transform.
Abstract: Introduction Preliminaries and notation The what, why, and how of wavelets The continuous wavelet transform Discrete wavelet transforms: Frames Time-frequency density and orthonormal bases Orthonormal bases of wavelets and multiresolutional analysis Orthonormal bases of compactly supported wavelets More about the regularity of compactly supported wavelets Symmetry for compactly supported wavelet bases Characterization of functional spaces by means of wavelets Generalizations and tricks for orthonormal wavelet bases References Indexes.

14,157 citations

Book ChapterDOI
01 Jan 1995
TL;DR: A reconstruction subject to far weaker Gibbs phenomena than thresholding based De-Noising using the traditional orthogonal wavelet transform is produced.
Abstract: De-Noising with the traditional (orthogonal, maximally-decimated) wavelet transform sometimes exhibits visual artifacts; we attribute some of these—for example, Gibbs phenomena in the neighborhood of discontinuities—to the lack of translation invariance of the wavelet basis. One method to suppress such artifacts, termed “cycle spinning” by Coifman, is to “average out” the translation dependence. For a range of shifts, one shifts the data (right or left as the case may be), De-Noises the shifted data, and then unshifts the de-noised data. Doing this for each of a range of shifts, and averaging the several results so obtained, produces a reconstruction subject to far weaker Gibbs phenomena than thresholding based De-Noising using the traditional orthogonal wavelet transform.

1,888 citations


"A real algebra perspective on multi..." refers background in this paper

  • ...It has been observed in [14] that redundancy of wavelet frames has advantages for applications in signal denoising - if the data is redundant, then loosing some data during transmission does not necessarily affect the reconstruction of the original signal....

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Book
02 Oct 2009
TL;DR: The semidefinite programming methodology to solve the generalized problem of moments is presented and several applications of the GPM are described in detail (notably in optimization, probability, optimal control andmathematical finance).
Abstract: . From a theoretical viewpoint, the GPM has developments and impact in var-ious area of Mathematics like algebra, Fourier analysis, functional analysis, operator theory, probabilityand statistics, to cite a few. In addition, and despite its rather simple and short formulation, the GPMhas a large number of important applications in various fields like optimization, probability, mathematicalfinance, optimal control, control and signal processing, chemistry, cristallography, tomography, quantumcomputing, etc.In its full generality, the GPM is untractable numerically. However when K is a compact basic semi-algebraic set, and the functions involved are polynomials (and in some cases piecewise polynomials orrational functions), then the situation is much nicer. Indeed, one can define a systematic numerical schemebased on a hierarchy of semidefinite programs, which provides a monotone sequence that converges tothe optimal value of the GPM. (A semidefinite program is a convex optimization problem which up toarbitrary fixed precision, can be solved in polynomial time.) Sometimes finite convergence may evenocccur.In the talk, we will present the semidefinite programming methodology to solve the GPM and describein detail several applications of the GPM (notably in optimization, probability, optimal control andmathematical finance).R´ef´erences[1] J.B. Lasserre, Moments, Positive Polynomials and their Applications, Imperial College Press, inpress.[2] J.B. Lasserre, A Semidefinite programming approach to the generalized problem of moments,Math. Prog. 112 (2008), pp. 65–92.

1,020 citations


"A real algebra perspective on multi..." refers background in this paper

  • ...This establishes a connection between constructions of tight wavelet frames and moment problems, see [24, 30, 31] for details....

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Journal ArticleDOI
TL;DR: A new interpolatory subdivision scheme for surface design is presented that is designed for a general triangulation of control points and has a tension parameter that provides design flexibility.
Abstract: A new interpolatory subdivision scheme for surface design is presented. The new scheme is designed for a general triangulation of control points and has a tension parameter that provides design flexibility. The resulting limit surface is C1 for a specified range of the tension parameter, with a few exceptions. Application of the butterfly scheme and the role of the tension parameter are demonstrated by several examples.

872 citations


"A real algebra perspective on multi..." refers background or methods in this paper

  • ...We illustrate this method on the example of the so-called butterfly scheme from [19]....

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  • ...The butterfly scheme describes an interpolatory subdivision scheme that generates a smooth regular surface interpolating a given set of points [19]....

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  • ...13 illustrates the advantage of the representation in (32) for the butterfly scheme [19], an interpolatory subdivision method with the corresponding mask p ∈ C[T ] of a larger support, some of whose coefficients are negative....

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Journal ArticleDOI
TL;DR: Wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames, are discussed and it is shown how they can be used for systematic constructions of spline, pseudo-spline tight frames, and symmetric bi-frames with short supports and high approximation orders.

764 citations


"A real algebra perspective on multi..." refers background or methods in this paper

  • ...The starting point of our study is the so-called Unitary Extension Principle (UEP) from [33], a special case of the above mentioned characterizations in [8, 9, 16]....

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  • ...We list some existing constructions of compactly supported MRA wavelet tight frames of L2(R ) [7, 10, 16, 23, 29, 33, 38] that employ the Unitary Extension Principle....

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  • ...1 Formulations of UEP in wavelet frame literature Most formulations of the UEP are given in terms of identities for trigonometric polynomials, see [16, 33]....

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  • ...1 is a wavelet tight frame of L2(R ), see [16, 33]....

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