p-exponent and p-leaders, Part II: Multifractal Analysis. Relations to Detrended Fluctuation Analysis

Multifractal analysis studies signals, functions, images or fields via the fluctuations of their local regularity along time or space, which capture crucial features of their temporal/spatial dynamics. It has become a standard signal and image processing tool and is commonly used in numerous applications of different natures. In its common formulation, it relies on the Holder exponent as a measure of local regularity, which is by nature restricted to positive values and can hence be used for locally bounded functions only. In this contribution, it is proposed to replace the Holder exponent with a collection of novel exponents for measuring local regularity, the $p$-exponents. One of the major virtues of $p$-exponents is that they can potentially take negative values. The corresponding wavelet-based multiscale quantities, the $p$-leaders, are constructed and shown to permit the definition of a new multifractal formalism, yielding an accurate practical estimation of the multifractal properties of real-world data. Moreover, theoretical and practical connections to and comparisons against another multifractal formalism, referred to as multifractal detrended fluctuation analysis, are achieved. The performance of the proposed $p$-leader multifractal formalism is studied and compared to previous formalisms using synthetic multifractal signals and images, illustrating its theoretical and practical benefits. The present contribution is complemented by a companion article studying in depth the theoretical properties of $p$-exponents and the rich classification of local singularities it permits.

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

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 Holder 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.