arXiv subjects

Stéphane Jaffard

Publications and source records attributed to Stéphane Jaffard.

24 records · Page 2Linked to original sources

arXiv2012

Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures

Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities constructed upon the coefficients of hyperbolic wavelet decompositions. A multifractal analysis is introduced, that jointly accounts for scale invariance and anisotropy. Its properties are studied in depth.

math.FA↗

arXiv2012

Irregularities and Scaling in Signal and Image Processing: Multifractal Analysis

B. Mandelbrot gave a new birth to the notions of scale invariance, selfsimilarity and non-integer dimensions, gathering them as the founding corner-stones used to build up fractal geometry. The first purpose of the present contribution is to review and relate together these key notions, explore their interplay and show that they are different facets of a same intuition. Second, it will explain how these notions lead to the derivation of the mathematical tools underlying multifractal analysis. Third, it will reformulate these theoretical tools into a wavelet framework, hence enabling their better theoretical understanding as well as their efficient practical implementation. B. Mandelbrot used his concept of fractal geometry to analyze real-world applications of very different natures. As a tribute to his work, applications of various origins, and where multifractal analysis proved fruitful, are revisited to illustrate the theoretical developments proposed here.

math.FA↗

arXiv2012

Local Multifractal Analysis

We introduce a local multifractal formalism adapted to functions, measures or distributions which display multifractal characteristics that can change with time, or location. We develop this formalism in a general framework and we work out several examples of measures and functions where this setting is relevant.

math.CA↗

arXiv2012

Multivariate Davenport series

We consider series of the form $\sum a_n \{n\cdot x\}$, where $n\in\Z^{d}$ and $\{x\}$ is the sawtooth function. They are the natural multivariate extension of Davenport series. Their global (Sobolev) and pointwise regularity are studied and their multifractal properties are derived. Finally, we list some open problems which concern the study of these series.

math.NT↗

arXiv2010

Multifractal analysis of Lévy fields

We study the pointwise regularity properties of the Lévy fields introduced by T. Mori; these fields are the most natural generalization of Lévy processes to the multivariate setting. We determine their spectrum of singularities, and we show that their Hölder singularity sets satisfy a large intersection property in the sense of K. Falconer.

math.PR↗

arXiv2003

Random Wavelet Series: Theory and Applications

Random Wavelet Series form a class of random processes with multifractal properties. We give three applications of this construction. First, we synthesize a random function having any given spectrum of singularities satisfying some conditions (but including non-concave spectra). Second, these processes provide examples where the multifractal spectrum coincides with the spectrum of large deviations, and we show how to recover it numerically. Finally, particular cases of these processes satisfy a generalized selfsimilarity relation proposed in the theory of fully developed turbulence.

math-ph↗