Herwig Wendt

Wavelet leaders and bootstrap for multifractal analysis of images

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. Here, 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 performances 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. 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. Performances 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.

Multifractality Tests Using Bootstrapped Wavelet Leaders

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.

Wavelet Leader multifractal analysis for texture classification

Image classification often relies on texture characterization. Yet texture characterization has so far rarely been based on a true 2D multifractal analysis. Recently, a 2D wavelet Leader based multifractal formalism has been proposed. It allows to perform an accurate, complete and low computational and memory costs multifractal characterization of textures in images. This contribution describes the first application of such a formalism to a real large size (publicly available) image database, consisting of 25 classes of non traditional textures, with 40 high resolution images in each class. Multifractal attributes are estimated from each image and used as classification features within a standard k nearest neighbor classification procedure. The results reported here show that this Leader based multifractal analysis enables the effective discrimination of different textures, as performances in both classification scores and computational costs compare favorably against those of procedures previously proposed in the literature on the same database.

PURSUING AUTOMATED CLASSIFICATION OF HISTORIC PHOTOGRAPHIC PAPERS FROM RAKING LIGHT IMAGES

Abstract Surface texture is a critical feature in the manufacture, marketing, and use of photographic paper. Raking light reveals texture through a stark rendering of highlights and shadows. Though close-up raking light images effectively document surface features of photographic paper, the sheer number and diversity of textures used for historic papers prohibits efficient visual classification. This work provides evidence that automatic, computer-based classification of texture documented with raking light is feasible by demonstrating an encouraging degree of success sorting a set of 120 images made from samples of historic silver gelatin paper. Using this dataset, four university teams applied different image-processing strategies for automatic feature extraction and degree of similarity quantification. All four approaches successfully detected strong affinities and outliers built into the dataset. The creation and deployment of the algorithms was carried out by the teams without prior knowledge of the distributions of similarities and outliers. These results indicate that automatic classification of silver gelatin photographic paper based on close-up texture images is feasible and should be pursued. To encourage the development of other classification schemes, the 120-sample “training” dataset used in this work is available to other academic researchers at http://www.PaperTextureID.org.

Methodology for multifractal analysis of heart rate variability: From LF/HF ratio to wavelet leaders

The present contribution aims at proposing a comprehensive and tutorial introduction to the practical use of wavelet Leader based multifractal analysis to study heart rate variability. First, the theoretical background is recalled. Second, practical issues and pitfalls related to the selection of the scaling range or statistical orders, minimal regularity, parabolic approximation of spectrum and parameter estimation, are discussed. Third, multifractal analysis is connected explicitly to other standard characterizations of heart rate variability: (mono)fractal analysis, Hurst exponent, spectral analysis and the HF/LF ratio. This review is illustrated on real per partum fetal ECG data, collected at an academic French public hospital, for both healthy fetuses and fetuses suffering from acidosis.

Testing fractal connectivity in multivariate long memory processes

Within the framework of long memory multivariate processes, fractal connectivity is a particular model, in which the low frequencies (coarse scales) of the interspectrum of each pair of process components are determined by the autospectra of the components. The underlying intuition is that long memories in each components are likely to arise from a same and single mechanism. The present contribution aims at defining and characterizing a statistical procedure for testing actual fractal connectivity amongst data. The test is based on Fishers Z transform and Pearson correlation coefficient, and anchored in a wavelet framework. Its performance are analyzed theoretically and validated on synthetic data. Its usefulness is illustrated on the analysis of Internet traffic Packet and Byte count time series.

Bootstrap for Multifractal Analysis

Multifractal analysis, which mainly consists in estimating scaling exponents, has become a popular tool for empirical data analysis. Although widely used in different applications, the statistical performance and the reliability of the estimation procedures are still poorly known. Notably, little is known about confidence intervals, though they are of first importance in applications. The present work investigates the potential uses of bootstrap for multifractal estimation. Can bootstrap improve current estimation procedures or be used to obtain reliable confidence intervals? Comparing the statistical performance of different estimators, our major result is to show that bootstrap based procedures provide us both with accurate estimates and reliable confidence intervals. We believe that this brings substantial improvements to practical empirical multifractal analyses

On the impact of the number of vanishing moments on the dependence structures of compound Poisson motion and fractional Brownian motion in multifractal time

From a theoretical perspective, scale invariance, or simply scaling, can fruitfully be modeled with classes of multifractal stochastic processes, designed from positive multiplicative martingales (or cascades). From a practical perspective, scaling in real-world data is often analyzed by means of multiresolution quantities. The present contribution focuses on three different types of such multiresolution quantities, namely increment, wavelet and Leader coefficients, as well as on a specific multifractal processes, referred to as Infinitely Divisible Motions and fractional Brownian motion in multifractal time. It aims at studying, both analytically and by numerical simulations, the impact of varying the number of vanishing moments of the mother wavelet and the order of the increments on the decay rate of the (higher order) covariance functions of the (q-th power of the absolute values of these) multiresolution coefficients. The key result obtained here consist of the fact that, though it fastens the decay of the covariance functions, as is the case for fractional Brownian motions, increasing the number of vanishing moments of the mother wavelet or the order of the increments does not induce any faster decay for the (higher order) covariance functions

Wavelet decomposition of measures: Application to multifractal analysis of images

We show the relevance of multifractal analysis for some problems in image processing. We relate it to the standard question of the determination of correct function space settings. We show why a scale-invariant analysis, such as the one provided by wavelets, is pertinent for this purpose. Since a good setting for images is provided by spaces of measures, we give some insight into the problem of multifractal analysis of measures using wavelet techniques.

Bootstrap tests for the time constancy of multifractal attributes

On open and controversial issue in empirical data analysis is to decide whether scaling and multifractal properties observed in empirical data actually exist, or whether they are induced by intricate non stationarities. To contribute to answering this question, we propose a procedure aiming at testing the constancy along time of multifractal attributes estimated over adjacent non overlapping time windows. The procedure is based on non parametric bootstrap resampling and on wavelet Leader estimations for the multifractal parameters.lt is shown, by means of numerical simulations on synthetic multifractal processes, that the proposed procedure is reliable and powerful for discriminating true scaling behavior against non stationarities. We end up with a practical procedure that can be applied to a single finite length observation of data with unknown statistical properties.