Wavelet Transform Modulus Maxima Research Articles

Linear and nonlinear characterization of surfaces from a laser beam melt ablation process

We apply linear and nonlinear methods to study the properties of surfaces generated by a laser beam melt ablation process. As a result we present a characterization and ordering of the surfaces depending on the adjusted process parameters. Our findings give some insight into the performance of two widely applied multifractal analysis methods—the detrended fluctuation analysis and the wavelet transform modulus maxima method—on short real world data.

Numerical methods for the estimation of multifractal singularity spectra on sampled data: A comparative study

Physical variables in scale invariant systems often show chaotic, turbulent-like behavior, commonly associated to the existence of an underlying fractal or multifractal structure. However, the assessment of multifractality over experimental, discretized data requires of appropriate methods and to establish criteria to measure the confidence degree on the estimates. In this paper we have evaluated the quality of different techniques used for multifractal analysis. We have tested four different techniques: the moment (M) method, the wavelet transform modulus maxima (WTMM) method, the gradient modulus wavelet projection (GMWP) method and the gradient histogram (GH) method, which are used to estimate the singularity spectra of multifractal signals. The test consists in analyzing synthetic multifractal 1D signals with given multifractal spectrum. We have compared the results, studying the sensibility of each method to the length of the series, size of the ensemble and type of spectrum. Our results show that GMWP method is the one attaining the best performance, providing reliable estimates which can be improved when the statistics is increased. All the other methods are affected by problems such as the linearization of the right tail of the spectrum, and some of them are very demanding in data.

An application of the modulus maxima distribution in machinery condition monitoring

PurposeTo investigate wavelet modulus maxima distribution (MMD) in machinery condition monitoring and extract a parameter that can give a quantitative description of machinery‐operating status.Design/methodology/approachSignal decomposition technique is applied to extract gear motion signal and then wavelet transform modulus maxima are utilized to define fault growth parameter (FGP).FindingsMMD were proposed and the distribution used to derive an EWMA statistic representing machinery fault growth. A comparison with other research works indicates better performance of this parameter.Practical implicationsThis paper presents an innovative scheme for the machinery condition monitoring, on the basis of wavelet modulus maxima representation. The definition of MMD can be utilized to derive a parameter that describes the operating status of machinery. This parameter is load‐independent so that it demonstrates better performance when compared with other research works. Further, the MMD may be treated as input of condition classification system in the future work.Originality/valueThe idea for this paper stems from wavelet modulus maxima representation, whilst the application in vibration signal analysis is new. It was found that, by applying this approach, the occurrence of failure is correctly identified and the proposed EWMA FGP is independent of the load applied, which is a very important property in machinery condition monitoring and fault detection.

Wavelet-based multiscale resolution analysis of real and simulated time-series of earthquakes

SUMMARY This study introduces a new approach (based on the Continuous Wavelet Transform Modulus Maxima method) to describe qualitatively and quantitatively the complex temporal patterns of seismicity, their multifractal and clustering properties in particular. Firstly, we analyse the temporal characteristics of intermediate-depth seismic activity (M ≥ 2.6 events) in the Vrancea region, Romania, from 1974 to 2002. The second case studied is the shallow, crustal seismicity (M ≥ 1.5 events), which occurred from 1976 to 1995 in a relatively large region surrounding the epicentre of the 1995 Kobe earthquake (Mw = 6.9). In both cases we have declustered the earthquake catalogue and selected only the events with M ≥ Mc (where Mc is the magnitude of completeness) before analysis. The results obtained in the case of the Vrancea region show that for a relatively large range of scales, the process is nearly monofractal and random (does not display correlations). For the second case, two scaling regions can be readily noticed. At small scales (i.e. hours to days) the series display multifractal behaviour, while at larger scales (days

Multiscale corner detection of contour images using wavelet transform

A new multiscale corner detection method is proposed based on dyadic wavelet transform (WT) of the orientation function of a contour image. As the decomposition of the dyadic WT is complete and its scales are sparse, all the scales are defined as natural scales for corner detection. The points that are wavelet transform modulus maxima (WTMM) at different scales are taken as corner candidates. For each corner candidate, the sum of the corresponding normalized WTMM at all the natural scales is used as significance measure of the "cornerness". The utilization of the complete information makes the performance of the proposed detector independent to the type of input images. The decomposition scales of the WT are restricted by the contour length, which makes the algorithm adaptable for both long contours and short contours. Both subjective and objective evaluation illustrate better performance of the proposed corner detector compared to the conventional methods.

Scaling behavior of high resolution temporal rainfall: New insights from a wavelet-based cumulant analysis

We revisit the multifractal analysis of high resolution temporal rainfall using the wavelet transform modulus maxima (WTMM) method. Specifically, we employ a cumulant analysis of the logarithm of the WTMM coefficients to estimate the scaling exponent spectrum τ ( q ) and the spectrum of singularities D ( h ) . We document that rainfall intensity fluctuations exhibit multifractality from scales of the order of 4–5 minutes up to the storm-pulse duration of 1–2 hours. We also establish long-range dependence consistent with that of a multiplicative cascade.

Fractional Fourier Transform of Cantor Sets

A new kind of multifractal is constructed by fractional Fouriertransform of Cantor sets. The wavelet transform modulus maxima methodis applied to calculate the singularity spectrum under an operationaldefinition of multifractal. In particular, an analysing procedure todetermine the spectrum is suggested for practice. Nonanalyticities ofsingularity spectra or phase transitions are discovered, which areinterpreted as some indications on the range of Boltzmann temperatureq, on which the scaling relation of partition function holds.

Identification of coding and non-coding sequences using local Holder exponent formalism

Accurate prediction of genes in genomes has always been a challenging task for bioinformaticians and computational biologists. The discovery of existence of distinct scaling relations in coding and non-coding sequences has led to new perspectives in the understanding of the DNA sequences. This has motivated us to exploit the differences in the local singularity distributions for characterization and classification of coding and non-coding sequences. The local singularity density distribution in the coding and non-coding sequences of four genomes was first estimated using the wavelet transform modulus maxima methodology. Support vector machines classifier was then trained with the extracted features. The trained classifier is able to provide an average test accuracy of 97.7%. The local singularity features in a DNA sequence can be exploited for successful identification of coding and non-coding sequences. Available on request from bd.kulkarni@ncl.res.in.

Effect of independent component analysis on multifractality of EEG during visual-motor task

In visual-motor task, blinks, muscle and other short-term artifacts may corrupt the correct determination of the fractal properties of the brain signals that reflect task performance. We tested whether independent component analysis (ICA) is a necessary preprocessing tool to determine fractal properties of amplitudes of decomposed EEG records in theta, alpha, beta and gamma oscillations. The subjects were required to track a moving spot on a city map displayed on a computer screen by pushing forward a joystick in two experimental conditions: the spot moved in steps either with constant or fractal time intervals. The wavelet transform modulus maxima (WTMM) method was applied to estimate the local fractal dimensions D(h) for recovered EEG via reduced independent components (ICs) localized inside and outside the brain. The measure of fractality, i.e. the Hoelder exponent h D max, was statistically estimated among experimental conditions. We established multifractality for extracted IC per se, specific for filtered oscillations in both experimental conditions: long-range correlation for theta and alpha, and anticorrelation for beta and gamma. Similar results were obtained for filtered versions of recovered EEG. Multifractal scaling, specific for lower and higher EEG oscillations, proved to be very stable intrinsic feature for the activity of large brain areas. The external events (task conditions) and the extended number of ICs, including those at the boundary line of the brain, did not have influence upon the scaling, although their effects might be statistically different for a given filtered oscillation.

Multifractal analysis of particle-fluid system in a circulating fluidized bed

In this paper, multifractal analysis together with wavelet transform modulus maxima (WTMM) method is used to analyze the fluctuating signals of circulating fluidized bed (CFB). Singularity spectrum shows that the gas-particle flow in CFB has multifractal character. Motion behavior of the particle-fluid system of CFB can be described by singularity spectrum. Intermittency index can be used to determine the transition of flow regime from fast fluidization to pneumatic conveying.

Multifractal characterization of blood pressure dynamics: stress-induced phenomena

We investigate the scaling features of blood pressure dynamics in healthy rats by means of the wavelet transform modulus maxima method. We discuss how stress affects the phenomenon of multifractality in the cardiovascular dynamics. Typical reactions to stress are considered, and distinctions in the stress-induced effects for male and female rats are reported.

Lacunarity of fractal superlattices: a remote estimation using wavelets

The lacunarity provides a useful parameter for describing the distribution of gap sizes in discrete self-similar (fractal) superlattices and is used in addition to the similarity dimension to describe fractals. We show here that lacunarity, as well as the similarity dimension, can be remotely estimated from the wavelet analysis of superlattices impulse response. As a matter of fact, the skeleton - the set of wavelet-transform modulus-maxima - of the reflected signal overlaps two hierarchical structures in the time-scale domain: such that one allows the direct remote extraction of the similarity dimension, while the other may provide an accurate estimation of the lacunarity of the interrogated superlattice. Criteria for the choice of the mother wavelet are established for impulse response corrupted by additive Gaussian white noise.

CONTINUOUS WAVELET TRANSFORM MODULUS MAXIMA ANALYSIS OF THE ELECTROCARDIOGRAM: BEAT CHARACTERISATION AND BEAT-TO-BEAT MEASUREMENT

The problem of automatic beat recognition in the ECG is tackled using continuous wavelet transform modulus maxima (CWTMM). Features within a variety of ECG signals can be shown to correspond to various morphologies in the CWTMM domain. This domain has an easy interpretation and offers a useful tool for the automatic characterization of the different components observed in the ECG in health and disease. As an application of this enhanced time-frequency analysis technique for ECG signals, an R-wave detector is developed and tested using patient signals recorded in the Coronary Care Unit of the Royal Infirmary of Edinburgh (attaining a sensitivity of 99.53% and a positive predictive value of 99.73%) and with the MIT/BIH database (attaining a sensitivity of 99.70% and a positive predictive value of 99.68%).

In Situ Selectivity and Thickness Monitoring During Selective Silicon Epitaxy Using Quadrupole Mass Spectrometry and Wavelets

This work reports on a novel in situ sensing technique for monitoring the thickness of selectively grown Si epitaxial layers. The technique can be extended to detect selectivity loss when Si nuclei begin to appear on the insulator surface. In this technique, a quadruple mass spectrometer (QMS) monitors the ionized molecular hydrogen (H/sub 2//sup +/) signal, which is a by-product of the chemical-vapor deposition process. The thickness of the epitaxial layer is determined by evaluating the area under the hydrogen signal. We have deliberately used silane (SiH/sub 4/) without HCl or Cl/sub 2/ to achieve both nonselective and selective depositions. We also show that the amount of hydrogen produced by the deposition process is a strong function of the exposed Si area on the wafer and the effect can be accurately monitored by QMS. This finding was exploited to develop an in situ sensing method to detect the selectivity loss. When selectivity is lost, Si nuclei begin to form on the insulator surface increasing the effective Si area on the wafer. Consequently, the rate of hydrogen production increases rapidly as nuclei coalesce, resulting in a distinct change in the functional form of the hydrogen signal. The hydrogen signal was analyzed using an automatic edge detection procedure based on the wavelet transform modulus maxima representation. The technique facilitated the determination of selective film thickness from the time-integrated hydrogen (H/sub 2//sup +/) signal. To the authors' knowledge, This work represents one of the first applications of wavelets to in situ process monitoring and fault detection in semiconductor manufacturing. The authors expect the methodology presented in This work to be readily transferable to other selective deposition processes, including those that utilize dichlorosilane and disilane since hydrogen is a by-product of those processes as well.

Multifractal characterization of blood pressure dynamics: stress-induced phenomena

We investigate the scaling features of blood pressure dynamics in healthy rats by means of the wavelet transform modulus maxima method. We discuss how stress affects the phenomenon of multifractality in the cardiovascular dynamics. Typical reactions to stress are considered, and distinctions in the stress-induced effects for male and female rats are reported. 2004 Elsevier Ltd. All rights reserved.

Influence of the atomic force microscope tip on the multifractal analysis of rough surfaces

In this paper, the influence of atomic force microscope tip on the multifractal analysis of rough surfaces is discussed. This analysis is based on two methods, i.e. on the correlation function method and the wavelet transform modulus maxima method. The principles of both methods are briefly described. Both methods are applied to simulated rough surfaces (simulation is performed by the spectral synthesis method). It is shown that the finite dimensions of the microscope tip misrepresent the values of the quantities expressing the multifractal analysis of rough surfaces within both the methods. Thus, it was concretely shown that the influence of the finite dimensions of the microscope tip changed mono-fractal properties of simulated rough surface to multifractal ones. Further, it is shown that a surface reconstruction method developed for removing the negative influence of the microscope tip does not improve the results obtained in a substantial way. The theoretical procedures concerning both the methods, i.e. the correlation function method and the wavelet transform modulus maxima method, are illustrated for the multifractal analysis of randomly rough gallium arsenide surfaces prepared by means of the thermal oxidation of smooth gallium arsenide surfaces and subsequent dissolution of the oxide films.

Scale-adaptive detection and local characterization of edges based on wavelet transform

In this paper, we present an edge detection and characterization method based on wavelet transform. This method relies on a modelization of contours as smoothed singularities of three particular types (transition, peak and line). Using the wavelet transform modulus maxima lines of the edge models, position and descriptive parameters of each edge point can be inferred. Indeed, on the one hand, the proposed algorithm allows to detect and locate edges at a locally adapted scale. On the other hand, it is able to identify the type of each detected edge point and to measure both its amplitude and smoothness degree. The latter parameters represent, respectively, the contrast and the blur level of the edge point. Evaluation of the method is performed on both synthetic and real images. Synthetic data are used to investigate the influence of different factors and the sensitivity to noise, whereas real images allow to highlight the performance and interests of the method. In particular, we point out that the measured smoothness degree provides a cue to recover depth from defocused images or a cue to diffusion measurements in images of cloud structures. Moreover, from an indoor scene, we demonstrate the relevance of type identification for segmentation purposes.

Generalizing the wavelet-based multifractal formalism to random vector fields: application to three-dimensional turbulence velocity and vorticity data.

We use singular value decomposition techniques to generalize the wavelet transform modulus maxima method to the multifractal analysis of vector-valued random fields. The method is calibrated on synthetic multifractal 2D vector measures and monofractal 3D fractional Brownian vector fields. We report the results of some application to the velocity and vorticity fields issued from 3D isotropic turbulence simulations. This study reveals the existence of an intimate relationship between the singularity spectra of these two vector fields which are found significantly more intermittent than previously estimated from longitudinal and transverse velocity increment statistics.

Multifractal spectrum of a laser beam melt ablation process

The surface profiles resulting from a laser beam melt ablation process have been quantified on the basis of the multifractal formalism. The multifractal spectra for several parameters sets, obtained with the wavelet transform modulus maxima method, were found to change towards higher Hölder exponents as the system undergoes a regime transition from medium to high ablation rates. This is consistent with the experimental observation of pattern formation at high ablation rates.

Multifractal structure of turbulence in the magnetospheric cusp

Abstract. Magnetospheric cusps are regions which are characterized by highly turbulent plasma. We have used Polar magnetic field data to study the structure of turbulence in the cusp region. The wavelet transform modulus maxima method (WTMM) has been applied to estimate the scaling exponent of the partition function and singularity spectra. Their features are similar to those found in the nonlinear multifractal systems. We have found that the scaling exponent does not allow one to conclude which intermittency model fits the experiment better. However, the singularity spectra reveal that different models can be ascribed to turbulence observed under various IMF conditions. For northward IMF conditions the turbulence is consistent with the multifractal p-model of fully developed fluid turbulence. For southward IMF experimental data agree with the model of non-fully developed Kolmogorov-like fluid turbulence.