No abstract

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We study the Gaussian random elds indexed by R d whose covariance is de ned in all generality as the parametrix of an elliptic pseudo-di erential operator with minimal regularity asumption on the symbol. We construct new wavelet bases adapted to these operators the decomposition of the eld on this corresponding basis yields its iterated logarithm law and its uniform modulus of continuity. We also characterize the local scalings of the eld in term of the properties of the principal symbol of the pseudodi erential operator. Similar results are obtained for the Multi-Fractional Brownian Motion.

Multifractal analysis is considered a promising tool for image processing, notably for texture characterization. However, practical operational estimation procedures based on a theoretically well established multifractal analysis are still lacking for image (as opposed to signal) processing. In the present contribution, a wavelet Leader based multifractal analysis, known to be theoretically strongly grounded, is described and assessed for 2D functions (images). By means of Monte Carlo simulations conducted over both self-similar and multiplicative cascade synthetic images, it is shown here to benefit from much better practical estimation performance than those obtained from a 2D discrete wavelet transform coefficient analysis. Furthermore, this is complemented by the original analysis and design of procedures aiming at practically assessing and handling the theoretical function space embedding requirements faced by multifractal analysis and certain image analysis methods. In addition, a bootstrap based statistical approach developed in the wavelet domain is proposed and shown to enable the practical computation of accurate confidence intervals for multifractal attributes from a given image. It is based on an original joint time and scale block non parametric bootstrap scheme. Performance are assessed by Monte Carlo simulations. Finally, the use and relevance of the proposed wavelet Leader and bootstrap based tools are illustrated at work on real-world images.