Wavelet techniques for pointwise regularity

Jaffard, Stéphane

doi:10.5802/afst.1111

Cited by 37 publications

(52 citation statements)

References 28 publications

(33 reference statements)

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“…It follows from (C.4) that the scalej ofλ satisfies j (1 − ε ) ≤j ≤ j . Therefore the corresponding p 0 -leader Remark that any ε > 0 can be written Cε , thus h p0,s (x 0 ) ≤ h p0 + s. Note that (C.1) implies that h p0 (x 0 ) ≥ h p0 , and the lower bound h p0,s ≥ h p0 + s then follows from general results on fractional integration, see [31,40]. We have thus obtained that X has a canonical singularity of exponent h p0 at x 0 .…”

Section: Proof Of Propositionmentioning

confidence: 88%

“…In this context, as a possible alternative to (fractional) integration, we propose the use of p-exponents, which potentially take negative values and hence permit to characterize negative local regularity. Though introduced in the theoretical context of PDEs as early as 1961 for p > 1 by Calderón and Zygmund [28], p-exponents were not used in signal processing until the 2000s when their wavelet characterization was proposed [29,30,31]. In the present contribution, we study which information on the local behavior of a function near a singularity is supplied by the knowledge of the collection of p-exponents.…”

Section: Introductionmentioning

confidence: 95%

“…When η X (p) > 0 and r ψ > h p (x 0 ), the p-exponent h p (x 0 ) defined in (6) can be recovered from p-leaders [31,26,29]:…”

Section: P-exponent Regularity Characterization With P-leadersmentioning

confidence: 99%

“…Essentially, this means that, as regards pointwise regularity, taking a fractional integral of order s is equivalent to multiplying the wavelet coefficients by 2 js , see [31,40] for a mathematical justification of this heuristic.…”

Section: Canonical Vs Oscillating Singularities 421 Fractional Inmentioning

confidence: 99%

“…This notion was introduced by A. Calderón and A. Zygmund [28] and has recently been put forward in the mathematical literature in [29] (see also [31,26]). It permits the definition of a collection of p-exponents to measure pointwise regularity.…”

Section: P-exponent Regularitymentioning

confidence: 99%

See 4 more Smart Citations

p-exponent and p-leaders, Part I: Negative pointwise regularity

Jaffard

Mélot

Leonarduzzi

et al. 2016

Physica A: Statistical Mechanics and its Applications

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Multifractal analysis aims to characterize signals, functions, images or fields, via the fluctuations of their local regularity along time or space, hence capturing crucial features of their temporal/spatial dynamics. Multifractal analysis is becoming a standard tool in signal and image processing, and is nowadays widely used in numerous applications of different natures. Its common formulation relies on the measure of local regularity via the Hölder exponent, by nature restricted to positive values, and thus to locally bounded functions or signals. It is here proposed to base the quantification of local regularity on p-exponents, a novel local regularity measure potentially taking negative values. First, the theoretical properties of p-exponents are studied in detail. Second, wavelet-based multiscale quantities, the p-leaders, are constructed and shown to permit accurate practical estimation of p-exponents. Exploiting the potential dependence with p, it is also shown how the collection of p-exponents enriches the classification of locally singular behaviors in functions, signals or images. The present contribution is complemented by a companion article developing the p-leader based multifractal formalism associated to p-exponents.

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Section: Proof Of Propositionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 95%

“…When η X (p) > 0 and r ψ > h p (x 0 ), the p-exponent h p (x 0 ) defined in (6) can be recovered from p-leaders [31,26,29]:…”

Section: P-exponent Regularity Characterization With P-leadersmentioning

confidence: 99%

Section: Canonical Vs Oscillating Singularities 421 Fractional Inmentioning

confidence: 99%

Section: P-exponent Regularitymentioning

confidence: 99%

See 3 more Smart Citations

p-exponent and p-leaders, Part I: Negative pointwise regularity

Jaffard

Mélot

Leonarduzzi

et al. 2016

Physica A: Statistical Mechanics and its Applications

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Local theory for 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type

Saka

2023

Mathematische Nachrichten

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Multifractal Analysis of Images: New Connexions Between Analysis and Geometry

Heurteaux

Jaffard

Imaging for Detection and Identification

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Abstract. Natural images can be modelled as patchworks of homogeneous textures with rough contours. The following stages play a key role in their analysis:-Separation of each component -Characterization of the corresponding textures -Determination of the geometric properties of their contours. Multifractal analysis proposes to classify functions by using as relevant parameters the dimensions of their sets of singularities. This framework can be used as a classification tool in the last two steps enumerated above. Several variants of multifractal analysis were introduced, depending on the notion of singularity which is used. We describe the variants based on Hölder and L p regularity, and we apply these notions to the study of functions of bounded variation (indeed the BV setting is a standard functional assumption for modelling images, which is currently used in the first step for instance). We also develop a multifractal analysis adapted to contours, where the regularity exponent associated with points of the boundary is based on an accessibility condition. Its purpose is to supply classification tools for domains with fractal boundaries.

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Wavelet techniques for pointwise regularity

Cited by 37 publications

References 28 publications

p-exponent and p-leaders, Part I: Negative pointwise regularity

p-exponent and p-leaders, Part I: Negative pointwise regularity

Local theory for 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type

Multifractal Analysis of Images: New Connexions Between Analysis and Geometry

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