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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.

We develop a discretization and computational procedures for approximation of the action of Fourier integral operators the canonical relations of which are graphs. Such operators appear, for instance, in the formulation of imaging and inverse scattering of seismic reflection data. Our discretization and algorithms are based on a multiscale low-rank expansion of the action of Fourier integral operators using the dyadic parabolic decomposition of phase space and on explicit constructions of low-rank separated representations using prolate spheroidal wave functions, which directly reflect the geometry of such operators. The discretization and computational procedures connect to the discrete almost symmetric wave packet transform. Numerical wave propagation and imaging examples illustrate our computational procedures. F. ANDERSSON, M. V. DE HOOP, AND H. WENDT 3d−1 2 log(N )), or O(D N d log(N )) if D is the number of significant tiles in the dyadic parabolic decomposition of u, valid in arbitrary finite dimension d. Our separated representation is expressed in terms of geometric attributes of the canonical relation of the FIO. We make use of prolate spheroidal wave functions (PSWFs) in connection with the dyadic parabolic decomposition, while the propagation of singularities or canonical transformation is accounted for via an unequally spaced FFT (USFFT). The use of PSWFs was motivated by the work of Beylkin and Sandberg [7] and the proposition of an efficient algorithm for their numerical evaluation by Xiao, Rokhlin, and Yarvin [62]. We note that it is also possible to obtain low-rank separated representations of the complex exponential in (1.1) purely numerically at the cost of losing the explicit relationship with the geometry. The algorithm presented here can be applied to computing parametrices of hyperbolic evolution and wave equations; we show that then our approximation corresponds to the solution of the paraxial wave equation in curvilinear coordinates, MULTISCALE DISCRETE APPROXIMATION OF FIOs 113 i.e., directionally developed relative to the central wave vector. However, it also forms the basis of a computational procedure following the construction of weak solutions of Cauchy initial value problems for the wave equation if the medium is C 2,1 , in which, in addition, a Volterra equation needs to be solved (de Hoop et al. [18]).We derive our discretization from the (inverse) transform based on discrete almost symmetric wave packets [25]. The connection of our algorithm to discrete almost symmetric wave packets is important in imaging and inverse scattering applications, where the FIOs act on data (u in the above). The wave packets can aid in regularizing the data from a finite set of samples through sparse optimization (instead of standard interpolation, for example) [14,15,17,58]. Moreover, the mentioned connection enables multiscale imaging and, in the context of directional pointwise regularity analysis [1,30,32,33,34,35,42], the numerical estimation and study of propagation of scaling exponents by the ...

Abstract-Multifractal analysis, which mostly consists of measuring scaling exponents, is becoming a standard technique available in most empirical data analysis toolboxes. Making use of the most recent theoretical results, it is based here on the estimation of the cumulants of the log of the wavelet Leaders, an elaboration on the wavelet coefficients. These log-cumulants theoretically enable discrimination between mono-and multifractal processes, as well as between simple log-normal multifractal models and more advanced ones. The goal of the present contribution is to design nonparametric bootstrap hypothesis tests aiming at testing the nature of the multifractal properties of stochastic processes and empirical data. Bootstrap issues together with six declinations of test designs are analyzed. Their statistical performance (significances, powers, and p-values) are assessed and compared by means of Monte Carlo simulations performed on synthetic stochastic processes whose multifractal properties (and log-cumulants) are known theoretically a priori. We demonstrate that the joint use of wavelet Leaders, log-cumulants, and bootstrap procedures enable us to obtain a powerful tool for testing the multifractal properties of data. This tool is practically effective and can be applied to a single observation of data with finite length.