Herwig Wendt

Bootstrap for Empirical Multifractal Analysis

Multifractal analysis is becoming a standard statistical analysis technique. In signal processing, it mostly consists of estimating scaling exponents characterizing scale invariance properties. For practical purposes, confidence intervals in estimation and p values in hypothesis testing are of primary importance. In empirical multifractal analysis, the statistical performance of estimation or test procedures remain beyond analytical derivation because of the theoretically involved nature of multifractal processes. Therefore, the goal of this article is to show how non-parametric bootstrap approaches circumvent such limitations and yield procedures that exhibit satisfactory statistical performance and can hence be practically used on real-life data. Such tools are illustrated at work on the analysis of the multifractal properties of empirical hydrodynamic turbulence data.

A New Frequency Estimation Method for Equally and Unequally Spaced Data

Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature. In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples. The method relies on Kroneckers theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix. This matrix is generated by sequences approximating the underlying sum of complex exponentials. Unequally spaced sampling is accounted for through a proper choice of interpolation matrices. The resulting optimization problem is then cast in a form that is suitable for using the alternating direction method of multipliers (ADMM). The method can easily include either a nuclear norm or a finite rank constraint for limiting the number of complex exponentials. The usage of a finite rank constraint makes, in contrast to the nuclear norm constraint, the method heuristic in the sense that the problem is non-convex and convergence to a global minimum can not be guaranteed. However, we provide a large set of numerical experiments that indicate that usage of the finite rank constraint nevertheless makes the method converge to minima close to the global minimum for reasonably high signal to noise ratios, hence essentially yielding maximum-likelihood parameter estimates. Moreover, the method does not seem to be particularly sensitive to initialization and performs substantially better than standard subspace-based methods.

Multiscale Anisotropic Texture Analysis and Classification of Photographic Prints: Art scholarship meets image processing algorithms

Texture characterization of photographic prints can provide scholars with valuable information regarding photographers? aesthetic intentions and working practices. Currently, texture assessment is strictly based on the visual acuity of a range of scholars associated with collecting institutions, such as museum curators and conservators. Natural interindividual discrepancies, intraindividual variability, and the large size of collections present a pressing need for computerized and automated solutions for the texture characterization and classification of photographic prints. In the this article, this challenging image processing task is addressed using an anisotropic multiscale representation of texture, the hyperbolic wavelet transform (HWT), from which robust multiscale features are constructed. Cepstral distances aimed at ensuring balanced multiscale contributions are computed between pairs of images. The resulting large-size affinity matrix is then clustered using spectral clustering, followed by a Ward linkage procedure. For proof of concept, these procedures are first applied to a reference data set of historic photographic papers that combine several levels of similarity and second to a large data set of culturally valuable photographic prints held by the Museum of Modern Art in New York. The characterization and clustering results are interpreted in collaboration with art scholars with an aim toward developing new modes of art historical research and humanities-based collaboration.

Multiscale Analysis of Intensive Longitudinal Biomedical Signals and Its Clinical Applications

Recent advances in wearable and/or biomedical sensing technologies have made it possible to record very long-term, continuous biomedical signals, referred to as biomedical intensive longitudinal data (ILD). To link ILD to clinical applications, such as personalized healthcare and disease prevention, the development of robust and reliable data analysis techniques is considered important. In this review, we introduce multiscale analysis methods for and the applications to two types of intensive longitudinal biomedical signals, heart rate variability (HRV) and spontaneous physical activity (SPA) time series. It has been shown that these ILD have robust characteristics unique to various multiscale complex systems, and some parameters characterizing the multiscale complexity are in fact altered in pathological states, showing potential usability as a new type of ambient diagnostic and/or prognostic tools. For example, parameters characterizing increased intermittency of HRV are found to be potentially useful in detecting abnormality in the state of the autonomic nervous system, in particular the sympathetic hyperactivity, and intermittency parameters of SPA might also be useful in evaluating symptoms of psychiatric patients with depressive as well as manic episodes, all in the daily settings. Therefore, multiscale analysis might be a useful tool to extract information on clinical events occurring at multiple time scales during daily life and the underlying physiological control mechanisms from biomedical ILD.

MultiScale Wavelet p-Leader based Heart Rate Variability Analysis for Survival Probability Assessment in CHF Patients

A priori discrimination of high mortality risk amongst congestive heart failure patients constitutes an important clinical stake in cardiology and involves challenging analyses of the temporal dynamics of heart rate variability (HRV). The present contribution investigates the potential of a new multifractal formalism, constructed on wavelet p-leader coefficients, to help discrimination between survivor and non survivor patients. The formalism, applied to a high quality database of 108 patients collected in a Japanese hospital, enables to assess the existence of multifractal properties amongst congestive heart failure patients and to reveal significant differences in the multiscale properties of HRV between survivor and non survivor patients, for scales ranging from approximately 60 to 250 beats.

A Bridge Between Geometric Measure Theory and Signal Processing: Multifractal Analysis

We describe the main features of wavelet techniques in multifractal analysis, using wavelet bases both as a tool for analysis, and for synthesis. We focus on two promising developments: We introduce the quantile leader method, which allows to put into light nonconcave multifractal spectra; we also test recent extensions of multifractal techniques fitted to functions that are not locally bounded but only belong to an L q space (determination of the q-spectrum).

Bayesian Estimation of the Multifractality Parameter for Image Texture Using a Whittle Approximation

Texture characterization is a central element in many image processing applications. Multifractal analysis is a useful signal and image processing tool, yet, the accurate estimation of multifractal parameters for image texture remains a challenge. This is due in the main to the fact that current estimation procedures consist of performing linear regressions across frequency scales of the 2D dyadic wavelet transform, for which only a few such scales are computable for images. The strongly non-Gaussian nature of multifractal processes, combined with their complicated dependence structure, makes it difficult to develop suitable models for parameter estimation. Here, we propose a Bayesian procedure that addresses the difficulties in the estimation of the multifractality parameter. The originality of the procedure is threefold. The construction of a generic semiparametric statistical model for the logarithm of wavelet leaders; the formulation of Bayesian estimators that are associated with this model and the set of parameter values admitted by multifractal theory; the exploitation of a suitable Whittle approximation within the Bayesian model which enables the otherwise infeasible evaluation of the posterior distribution associated with the model. Performance is assessed numerically for several 2D multifractal processes, for several image sizes and a large range of process parameters. The procedure yields significant benefits over current benchmark estimators in terms of estimation performance and ability to discriminate between the two most commonly used classes of multifractal process models. The gains in performance are particularly pronounced for small image sizes, notably enabling for the first time the analysis of image patches as small as 64 × 64 pixels.

Local regularity for texture segmentation: Combining wavelet leaders and proximal minimization

Texture segmentation constitutes a classical yet crucial task in image processing. In many applications of very different natures (biomedical, geophysics,...) textures are naturally defined in terms of their local regularity fluctuations, which can be quantified as the variations of local Hölder exponents. Furthermore, such images are often naturally embedded in the class of piece-wise constant local regularity functions. The present contribution aims at proposing and assessing a segmentation procedure for this class of images. Its originality is twofold: First, local regularity is estimated using wavelet leaders, a novel multiresolution quantity recently introduced for multifractal analysis but barely used in local regularity measurement, comparisons against wavelet coefficient based estimation are conducted; Second, the challenging minimal partition problem underlying segmentation is convexified and conducted within a customized proximal framework. The estimation of the number of regions and their target regularity is obtained from a total-variation estimate that enables the actual use of proximal minimization for texture segmentation. Performance is assessed and illustrated on synthetic textures.

Multifractal analysis of self-similar processes

Scale invariance and multifractal analysis are nowadays widely used in applications. For modeling scale invariance in data, two classes of processes are classically in competition: self-similar processes and multiplicative cascades. They imply fundamentally different underlying (additive or multiplicative) mechanisms, hence the crucial practical need for data driven model selection. Such identification relies on properties often associated with the former: self-similarity, monofractality, linear scaling function, null c2 parameter. By performing a wavelet leader based analysis of the multifractal properties of a large variety of self-similar processes, the present work contributes to a better disentangling of these different properties, sometimes confused one with another. Also, it leads to the formulation of conjectures regarding the scaling and multifractal properties of self-similar processes.

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.