Wavelet methods for pointwise regularity and local oscillations of functions

Jaffard, Stéphane; Meyer, Yves

doi:10.1090/memo/0587

Cited by 203 publications

(242 citation statements)

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“…However, for the step singularity, we found that, as a → 0, SH ψ H(a, 0, (0,t 2 )) ∼ a 3/4 (slow decay), whereas for the delta singularity we found SH ψ ν p (a, s,t) ∼ a −1/4 (increase). This shows that the sensitivity on the singularity type is consistent with the wavelet analysis as presented in [10,11]. Recall, in fact, that if f ∈ L 2 (R) is uniformly Lipschitz α in a neighborhood of t andψ is a nice wavelet, the continuous wavelet transform of f satisfies Wψ f (a,t) ≤ C a α+1/2 , which shows that the decay is controlled by the regularity of f at t. This analysis extends to the case where f has a jump or delta singularity at t, corresponding to α = 0 and α = −1, respectively.…”

Section: General Singularitiessupporting

confidence: 88%

“…Indeed, an even finer analysis of the local regularity properties of f , expressed in terms of local Lipschitz regularity, can be performed using the wavelet transform. This shows that there is a precise correspondence between the Lipschitz exponent α of f at a point x 0 (where α measures the regularity type) and the asymptotic behaviour of W ψ f (a, x 0 ) as a → 0 (see [11,10]). This is in contrast with the traditional Fourier analysis which is only sensitive to global regularity properties and cannot be used to measure the regularity of a function f at a specific location.…”

Section: Introductionmentioning

confidence: 91%

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Analysis and Identification of Multidimensional Singularities Using the Continuous Shearlet Transform

Guo

Labate

2012

Shearlets

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In this chapter, we illustrate the properties of the continuous shearlet transform with respect to its ability to describe the set of singularities of multidimensional functions and distributions. This is of particular interest since singularities and other irregular structures typically carry the most essential information in multidimensional phenomena. Consider, for example, the edges of natural images or the moving fronts in the solutions of transport equations. In the following, we show that the continuous shearlet transform provides a precise geometrical characterization of the singularity sets of multidimensional functions and precisely characterizes the boundaries of 2D and 3D regions through its asymptotic decay at fine scales. These properties go far beyond the continuous wavelet transform and other classical methods, and set the groundwork for very competitive algorithms for edge detection and feature extraction of 2D and 3D data.

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Section: General Singularitiessupporting

confidence: 88%

Section: Introductionmentioning

confidence: 91%

Analysis and Identification of Multidimensional Singularities Using the Continuous Shearlet Transform

Guo

Labate

2012

Shearlets

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“…Usually, the wavelet analysis presents two main important features [4,6,18,23,28,41]: the wavelet transform as a time-frequency analysis tool, and the wavelet analysis as part of approximation and function space theory (see also [4,16,17,27] and references therein for another approach to the time-frequency analysis). The existent applications of wavelet methods in local analysis are very rich.…”

Section: Introductionmentioning

confidence: 99%

Tauberian Theorems for the Wavelet Transform

Vindas

Pilipović

Rakić

2010

J Fourier Anal Appl

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Abstract. We make a complete wavelet analysis of asymptotic properties of distributions. The study is carried out via Abelian and Tauberian type results, connecting the boundary asymptotic behavior of the wavelet transform with local and non-local quasiasymptotic properties of elements in the Schwartz class of tempered distributions. Our Tauberian theorems are full characterizations of such asymptotic properties. We also provide precise wavelet characterizations of the asymptotic behavior of elements in the dual of the space of highly time-frequency localized functions over the real line. For the use of the wavelet transform in local analysis, we study the problem of extensions of distributions initially defined on R \ {0} to R; in this extension problem, we explore the asymptotic properties of extensions of a distribution having a prescribed asymptotic behavior. Our results imply intrinsic properties of functions and measures as well, for example, we give a new proof of the classical Littlewood Tauberian theorem for power series.

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“…One should bear in mind that the above relation is an approximate case for which exact theorems exist [9]. In particular, we will restrict the scope of this paper to Hölder singularities for which the local and pointwise Hölder exponents are equal [10].…”

Section: Assessing Singular Behaviour With the Wavelet Transformationmentioning

confidence: 99%

Local Effective Hölder Exponent Estimation on the Wavelet Transform Maxima Tree

Struzik

1999

Fractals

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Abstract. We present a robust method of estimating an effective Hölder exponent locally at an arbitrary resolution. The method is motivated by the multiplicative cascade paradigm, and implemented on the hierarchy of singularities revealed with the wavelet transform modulus maxima tree. In addition, we illustrate the possibility of the direct estimation of the scaling spectrum of the effective Hölder exponent, and we link it to the established partition functions based multifractal formalism. We motivate both the local and the global multifractal analysis by showing examples of computer generated and real life time series. keywords multifractal analysis, wavelet transform, Hölder exponent

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Wavelet methods for pointwise regularity and local oscillations of functions

Cited by 203 publications

References 0 publications

Analysis and Identification of Multidimensional Singularities Using the Continuous Shearlet Transform

Analysis and Identification of Multidimensional Singularities Using the Continuous Shearlet Transform

Tauberian Theorems for the Wavelet Transform

Local Effective Hölder Exponent Estimation on the Wavelet Transform Maxima Tree

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