Topic
Wavelet transform modulus maxima method
About: Wavelet transform modulus maxima method is a research topic. Over the lifetime, 53 publications have been published within this topic receiving 10091 citations.
Papers published on a yearly basis
Papers
TL;DR: It is proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents.
Abstract: The mathematical characterization of singularities with Lipschitz exponents is reviewed. Theorems that estimate local Lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed. It is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their Lipschitz exponents. The wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately. The local frequency of such oscillations is measured from the wavelet transform modulus maxima. It has been shown numerically that one- and two-dimensional signals can be reconstructed, with a good approximation, from the local maxima of their wavelet transform modulus. As an application, an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales. In two dimensions, the wavelet transform maxima indicate the location of edges in images. >
4,354 citations
TL;DR: In this article, the authors developed a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA).
Abstract: We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.
3,591 citations
TL;DR: The wavelet decomposition is used to generalize the multifractal formalism to singular signals and applications to fully developed turbulence data and Brownian signals are reported.
Abstract: The wavelet decomposition is used to generalize the multifractal formalism to singular signals The singularity spectrum is directly determined from the scaling behavior of partition functions that are defined from the wavelet transform modulus maxima Illustrations on fractal signals with a recursive structure, eg, devil's staircases, are shown Applications to fully developed turbulence data and Brownian signals are reported
828 citations
TL;DR: It is demonstrated that the method, based on the wavelet-transform modulus-maxima representation, works in most situations and is likely to be the ground of a unified multifractal description of self-affine distributions.
Abstract: Several attempts have been made recently to generalize the multifractal formalism, originally introduced for singular measures, to fractal signals. We report on a systematic comparison between the structure-function approach, pioneered by Parisi and Frisch [in 2 Proceedings of the International School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, Amsterdam, 1985), p. 84] to account for the multifractal nature of fully developed turbulent signals, and an alternative method we have developed within the framework of the wavelet-transform analysis. We comment on the intrinsic limitations of the structure-function approach; this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases. We demonstrate that our method, based on the wavelet-transform modulus-maxima representation, works in most situations and is likely to be the ground of a unified multifractal description of self-affine distributions. Our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations.
601 citations
TL;DR: In this article, the wavelet transform modulus maxima method was used to analyze the fractal scaling properties of DNA sequences using wavelet transforms. And the wavelets played the role of generalized oscillating boxes to determine the singularity spectrum of the considered signal and thereby to achieve a complete multifractal analysis.
181 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2018 | 1 |
| 2015 | 3 |
| 2014 | 5 |
| 2013 | 2 |
| 2012 | 3 |