Questioning normality: A study of wavelet leaders distribution

In this paper we propose to use the Wavelet Leader (WL) transformation for studying trabecular bone patterns. Given an input image, its WL transformation is defined as the cross-channel-layer maximum pooling of an underlying wavelet transformation. WL inherits the advantage of the original wavelet transformation in capturing spatial-frequency statistics of texture images, while being more robust against scale and orientation thanks to the maximum pooling strategy. These properties make WL an attractive alternative to replace wavelet transformations which are used for trabecular analysis in previous studies. In particular, in this paper, after extracting wavelet leader descriptors from a trabecular texture patch, we feed them into two existing statistic texture characterization methods, namely the Gray Level Co-occurrence Matrix (GLCM) and the Gray Level Run Length Matrix (GLRLM). The most discriminative features, Energy of GLCM and Gray Level Non-Uniformity of GLRLM, are retained to distinguish two different populations between osteoporotic patients and control subjects. Receiver Operating Characteristics (ROC) curves are used to measure performance of classification. Experimental results on a recently released benchmark dataset show that WL significantly boosts the performance of baseline wavelet transformations by 5% in average.

Singularity spectrum analysis of sea clutter is the key technology of detecting radar for sea target, which can discover the dynamic mechanism of the sea surface theoretically. In this paper, based on wavelet leaders the time-varying singularity spectrum distribution of sea clutters is proposed, which introduces time information to the traditional singularity spectrum, and displays the time-varying characteristic of singularity spectrum analytically. In theory, by way of self-windowed fractal signal, we introduce the time information to the traditional singularity spectrum, and realize multifractal spectrum distribution of sea clutters. In algorithm, based on the wavelet leaders, we adapt the process of embodying chirp-type and cusp-type singularities, and obtain the time-varying singularity spectrum distribution of sea clutters by the Legendre transform of the time-varying scaling function. In practice, we analyze the classical multifractal modelrandom wavelet series and the real sea clutter data of continuous wave Doppler radar in level III sea state. Experiments indicate that (1) the time-varying singularity spectrum distribution based on wavelet leaders can trace the time-varying scale characteristic and display the time-varying singularity spectrum distribution of sea clutters; (2) the algorithm possesses good statistical convergence, low computational cost, and passive moment property. The time-varying singularity spectrum distribution based on wavelet leaders may serve as a reference sample for nonlinear dynamics and multifractal signal processing.

Pathological examination is important for an accurate diagnosis of Synovial Sarcoma (SS). It is the most common cancer of the soft tissues of the limb in adolescents and adults. In this work, SS was used to determine the discriminant singularity characteristics using Wavelet Leaders (WL). The most popular technique for measuring the discriminant singularity characteristics of an image signal is the Lipschitz Exponent (LE). The singularity measurement was based on LE function by taking a slope of logarithmic scale versus logarithmic Wavelet Transform Modulus Maxima (WTMM). Here, the presence of the singularity was measured using WTMM and WL by summing each color component of an image signal. The performance characteristics of the statistical discrimination are evaluated and compared with the non-parametric hypothesis using the Wilcoxon rank-sum test. The most important difference between WTMM and WL was analyzed using the Receiver Operating Characteristics (ROC) curve. From the experimental analysis that the WL method provides excellent discriminant singularities or discontinuities performance characteristics such as area, standard error, z-statistics, and p-values. Finally, the results of experiments have proven that a WL can express practical, precise, robust, and satisfactory performance in practice.

In Ben Slimane (Mediterr J Math, 13(4):1513–1533 (2016)), the second author proved that, generically in the Baire category sense, pairs of functions in \({B_{t_{1}}^{s_{1},\infty}(\mathbb{R}^m) \times B_{t_{2}}^{s_{2},\infty}(\mathbb{R}^m) }\), for \({s_{1} > \frac{m}{t_{1}}}\) and \({s_{2} > \frac{m}{t_{2}}}\), satisfy a mixed multifractal formalism based on wavelet leaders. In this paper, we extend this validity on \({(B_{t_{1}}^{s_{1},\infty}(\mathbb{R}^m) \cap C^{\gamma_{1}}(\mathbb{R}^m)) \times (B_{t_{2}}^{s_{2},\infty}(\mathbb{R}^m) \cap C^{\gamma_{2}}(\mathbb{R}^m)}\), for \({0 < \gamma_{1} < s_{1} < \frac{m}{t_{1}}}\) and \({0 < \gamma_{2} < s_{2} < \frac{m}{t_{2}}}\). The main change is that the wavelet coefficients of the saturating function which generates the residual \({G_\delta}\) set are not everywhere large enough and do not coincide everywhere with the wavelet leaders. Nevertheless, the computation of the wavelet leaders is done everywhere and allows to deduce both mixed spectra and mixed scaling function.

Singularity detection in human EEG signal using wavelet leaders

Mutually interacting components form complex systems and the outputs of these components are usually long-range cross-correlated. Using wavelet leaders, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call joint multifractal analysis based on wavelet leaders (MF-X-WL). We test the validity of the MF-X-WL method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. Both experiments indicate that MF-X-WL is capable to detect the cross correlations in synthetic data with acceptable estimating errors. We also apply the MF-X-WL method to the pairs of series from financial markets (returns and volatilities) and online worlds (online numbers of different genders and different societies) and find an intriguing joint multifractal behavior.

In this paper we present an extended version of Hilbert-Huang transform, namely arbitrary-order Hilbert spectral analysis, to characterize the scale-invariant properties of a time series directly in an amplitude-frequency space. We first show numerically that due to a nonlinear distortion, traditional methods require high-order harmonic components to represent nonlinear processes, except for the Hilbert-based method. This will lead to an artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus the power law, if it exists, is contaminated. We then compare the Hilbert method with structure functions (SF), detrended fluctuation analysis (DFA), and wavelet leader (WL) by analyzing fractional Brownian motion and synthesized multifractal time series. For the former simulation, we find that all methods provide comparable results. For the latter simulation, we perform simulations with an intermittent parameter μ=0.15. We find that the SF underestimates scaling exponent when q>3. The Hilbert method provides a slight underestimation when q>5. However, both DFA and WL overestimate the scaling exponents when q>5. It seems that Hilbert and DFA methods provide better singularity spectra than SF and WL. We finally apply all methods to a passive scalar (temperature) data obtained from a jet experiment with a Taylor's microscale Reynolds number Re(λ)≃250. Due to the presence of strong ramp-cliff structures, the SF fails to detect the power law behavior. For the traditional method, the ramp-cliff structure causes a serious artificial energy flux from the low-frequency (large scale) to the high-frequency (small scale) part. Thus DFA and WL underestimate the scaling exponents. However, the Hilbert method provides scaling exponents ξ(θ)(q) quite close to the one for longitudinal velocity, indicating a less intermittent passive scalar field than what was believed before.

Digital clinical histopathology is one of the crucial techniques for precise cancer cell diagnosing in modern medicine. The Synovial Sarcoma (SS) cancer cell patterns seem to be a spindle shaped cell (SSC) structure and it is very difficult to identify the exact oval shaped cell structure through pathologist’s eye perception. Meanwhile, there is necessitating for monitoring and securing the successful and effective image data processing in the the huge network data which is also a complex one. A field programmable Gate Array (FPGA) was regarded as a necessary one for this. In this work, based on FPGA a Cancer Cell classification is made for the regulation and execution. Hence, mathematically the SSC regularity structures and its discontinuities are measured by the holder exponent (HE) function. In this research work, HE values have been determined by Wavelet Transform Modulus Maxima (WTMM) and Wavelet Leader (WL) methods with basis function of Haar wavelet based on FPGA Processor. The quantitative parameters such as Mean of Asymptotic Discontinuity (MAD), Mean of Removable Discontinuity (MRD) and Number of Discontinuity Points (NDPs) have been considered to determine the prediction of discontinuity detection between WTMM and WL methods. With the help of receiver operating characteristics (ROC) curve, the significant difference of discontinuity detection performance between both the methods has been analyzed. From the experimental results, it is clear that the WL method is more practically feasible and it gives satisfactory performance, in terms of sensitivity and specificity percentage values, which are 80.56% and 59.46%, respectively in the blue color components of the SNR 20 dB noise image.

In this paper, we propose an approach for Medical image analysis to detect tumors and to distinguish between healthy and pathological tissue that are present in the brain and skin. Our analysis is based on wavelet and multifractal formalism. In this analysis, we calculated the best linear regression interval that gives good parameter values calculated from new multiresolution indicator, called the average wavelet coefficient, derived from the wavelet leaders. Two main contributions are brought up: first, we proposed a method for the estimation of multifractal features. Second, we revealed the potential of multifractal features to characterize tumor brain and skin melanoma. We analyzed, compared our estimator and simulated image against wavelet leaders.

Wavelet leaders and bootstrap for multifractal analysis of images

The multifractal framework relates the scaling properties of turbulence to its local regularity properties through a statistical description as a collection of local singularities. The multifractal properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the multifractal properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well-founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The wavelet leaders yield a new multifractal formalism which meets all these requests. This paper aims at describing it and at applying it to experimental turbulent velocity data. After a detailed discussion of the practical use of the wavelet leader based multifractal formalism, the following questions are carefully investigated: (1) What is the dependence of multifractal properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which multifractal model does correctly account for the observed multifractal properties? Results from several data set analysis are used to discuss the dependence of the computed multifractal properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulant exponents up to order 4 confirms that the log-normal model (with c2 = -0.025±0.002) correctly accounts for the universal multifractal properties of turbulent velocity.

Wavelet Leaders: A new method to estimate the multifractal singularity spectra

The aim of this paper is to prove that wavelet leaders allow to get very fine properties of the trajectories of the Brownian motion: we show that the three well-known behaviors of its oscillations, namely to be ordinary, rapid and slow, are also present in the behavior of the size of its wavelet leaders.

In this work we analyze the correlation of the appearing T-wave alternans (TWA) with the shape (range) of the multifractal spectra of the ECG signals obtained with wavelet leaders based methods. A non-null correlation between the TWA occurring and lesser values for the Holder exponent of maximum dimension has been observed, in agreement with the idea that the projection of TWA must have a non-despicable contribution in the higher scales of the multiresolution analysis of the signal. We are developing new algorithms capable to deal with non-concave multifractal spectra based in wavelet leaders projections, instead of existing methods that only gives concave spectra. These new algorithms will give a finer resolution of Holder component of the signal and we hope that this richer information about the sharpness of the signal will allow better estimations of the magnitude of TWA and a closer relation between multifractal spectra and TWA occurring.

The purpose of this work is to analyze the multifractal features of uterine Electromyography (EMG) signals for the progression of pregnancy in term condition and to differentiate term (period $> 37$ weeks of gestation) and preterm (period $\leq 37$ weeks of gestation) conditions using Wavelet Leaders (WL) algorithm. For this study, the signals recorded from the surface of abdomen during the second (T1 and P1) and third trimester (T2) are considered from an online database. The signals are preprocessed and multifractal analysis is applied to compute the multifractal spectrum. Three features such as minimum $(\alpha_{\min})$ , maximum $(\alpha_{\max})$ and peak $(\alpha_{0})$ singularity exponents are extracted from the multifractal spectrum for analyzing the signals in T1, T2 and P1 groups. It is observed that there is a shift in the spectrum with increase in the order of wavelet. $\alpha_{\min}$ and $\alpha_{\max}$ are able to differentiate signals in T1-P1 and T1- T2 groups respectively. $\alpha_{0}$ is found to be consistent and has statistical significance in discriminating signals in all the considered groups. Hence, it appears that these multifractal features can help in investigating the progressive changes in uterine muscle contractions during pregnancy and differentiates term and preterm conditions at an early stage.