Function Spaces Vs. Scaling Functions: Tools for Image Classification

Jaffard, Stéphane; Abry, Patrice; Roux, Stéphane

doi:10.1007/978-3-642-19604-1_1

Cited by 20 publications

(41 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…Indeed, if h min > 0, then X is a continuous function, whereas, if h min < 0, then X / ∈ L ∞ loc , see [27,26]. For numerous real world applications, the restriction h min > 0 constitutes a severe limitation, cf.…”

Section: Global Regularity and Wavelet Coefficientsmentioning

confidence: 99%

“…Wavelet coefficients can furthermore be used to assess whether X locally belongs to L p or not, i.e., to verify the condition that is a priori required for using the corresponding p-exponent [26,32,27]. Let S c (j, p) denote the wavelet structure function, defined as the space/time averages of the magnitude of wavelet coefficients raised to a positive power p > 0:…”

Section: Global Regularity and Wavelet Coefficientsmentioning

confidence: 99%

“…This is notably quasi-systematically the case for biomedical time series, (cf. e.g., [26,27] and references therein for reviews).…”

Section: Introductionmentioning

confidence: 99%

“…The use of fractional integration, of order tuned to the uniform regularity of data, has recently been proposed and embedded directly in the definition of wavelet leaders [25,26,27]. However, fractional integration yields the significant issue of relating the characterization of local regularity of integrated functions to those of the original functions, prior to integration.…”

Section: Introductionmentioning

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

View full text Add to dashboard Cite

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.

show abstract