Empirical Wavelet Transform
TL;DR: This paper presents a new approach to build adaptive wavelets, the main idea is to extract the different modes of a signal by designing an appropriate wavelet filter bank, which leads to a new wavelet transform, called the empirical wavelets transform.
Abstract: Some recent methods, like the empirical mode decomposition (EMD), propose to decompose a signal accordingly to its contained information. Even though its adaptability seems useful for many applications, the main issue with this approach is its lack of theory. This paper presents a new approach to build adaptive wavelets. The main idea is to extract the different modes of a signal by designing an appropriate wavelet filter bank. This construction leads us to a new wavelet transform, called the empirical wavelet transform. Many experiments are presented showing the usefulness of this method compared to the classic EMD.
Citations
More filters
TL;DR: This work proposes an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently and is a generalization of the classic Wiener filter into multiple, adaptive bands.
Abstract: During the late 1990s, Huang introduced the algorithm called Empirical Mode Decomposition, which is widely used today to recursively decompose a signal into different modes of unknown but separate spectral bands. EMD is known for limitations like sensitivity to noise and sampling. These limitations could only partially be addressed by more mathematical attempts to this decomposition problem, like synchrosqueezing, empirical wavelets or recursive variational decomposition. Here, we propose an entirely non-recursive variational mode decomposition model, where the modes are extracted concurrently. The model looks for an ensemble of modes and their respective center frequencies, such that the modes collectively reproduce the input signal, while each being smooth after demodulation into baseband. In Fourier domain, this corresponds to a narrow-band prior. We show important relations to Wiener filter denoising. Indeed, the proposed method is a generalization of the classic Wiener filter into multiple, adaptive bands. Our model provides a solution to the decomposition problem that is theoretically well founded and still easy to understand. The variational model is efficiently optimized using an alternating direction method of multipliers approach. Preliminary results show attractive performance with respect to existing mode decomposition models. In particular, our proposed model is much more robust to sampling and noise. Finally, we show promising practical decomposition results on a series of artificial and real data.
4,111 citations
Cites background or methods from "Empirical Wavelet Transform"
...…local minima/maxima in a signal, estimates lower/ upper envelopes by interpolation of these extrema, removes the average of the envelopes as “low-pass” centerline, thus isolating the high-frequency oscillations as “mode” of a signal, and continues recursively on the remaining “low-pass” centerline....
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...The variational model assesses the bandwidth of the modes as -norm, after shifting the Hilbert-complemented, analytic signal down into baseband by complex harmonic mixing....
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TL;DR: VMD is a newly developed technique for adaptive signal decomposition, which can non-recursively decompose a multi-component signal into a number of quasi-orthogonal intrinsic mode functions and shows that the multiple features can be better extracted with the VMD, simultaneously.
418 citations
TL;DR: In this article, the inner product operation of wavelet transform (WT) is verified by simulation and field experiments and the development process of WT based on inner product is concluded and the applications of major developments in rotating machinery fault diagnosis are also summarized.
387 citations
TL;DR: The biggest challenge in realization of health monitoring of large real-life structures is automated detection of damage out of the huge amount of very noisy data collected from dozens of sensors on a daily, weekly, and monthly basis.
Abstract: Signal processing is the key component of any vibration-based structural health monitoring (SHM). The goal of signal processing is to extract subtle changes in the vibration signals in order to detect, locate and quantify the damage and its severity in the structure. This paper presents a state-of-the-art review of recent articles on signal processing techniques for vibration-based SHM. The focus is on civil structures including buildings and bridges. The paper also presents new signal processing techniques proposed in the past few years as potential candidates for future SHM research. The biggest challenge in realization of health monitoring of large real-life structures is automated detection of damage out of the huge amount of very noisy data collected from dozens of sensors on a daily, weekly, and monthly basis. The new methodologies for on-line SHM should handle noisy data effectively, and be accurate, scalable, portable, and efficient computationally.
349 citations
Additional excerpts
...Gilles [45] introduced empirical wavelet trans-...
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TL;DR: A novel hybrid deep-learning wind speed prediction model, which combines the empirical wavelet transformation and two kinds of recurrent neural network, is proposed, which indicates that the proposed model has satisfactory performance in the high-precision wind speed Prediction.
340 citations
References
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TL;DR: In this paper, a new method for analysing nonlinear and nonstationary data has been developed, which is the key part of the method is the empirical mode decomposition method with which any complicated data set can be decoded.
Abstract: A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...
18,956 citations
"Empirical Wavelet Transform" refers methods in this paper
...In this paper, as we will use the Fourier formalism in Section III, we adopt the description used in [4] which is slightly different from the original used in [9]....
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...[9] to extract such IMFs is a pure algorithmic method....
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...[9] proposed an original method called Empirical Mode Decomposition (EMD) to decompose a signal into specific modes (we define the meaning of “mode” hereafter)....
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...In this paper, we follow the idea used in the Hilbert-Huang transform [9]....
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TL;DR: Adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals, and relies heavily on the remarkable orthogonality properties of the new libraries.
Abstract: Adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals. It permits efficient compression of a variety of signals, such as sound and images. The predefined libraries of modulated waveforms include orthogonal wavelet-packets and localized trigonometric functions, and have reasonably well-controlled time-frequency localization properties. The idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost. The method relies heavily on the remarkable orthogonality properties of the new libraries: all expansions in a given library conserve energy and are thus comparable. Several cost functionals are useful; one of the most attractive is Shannon entropy, which has a geometric interpretation in this context. >
3,307 citations
"Empirical Wavelet Transform" refers background in this paper
...We also want to mention the differences with local cosine (sine) bases [1], [2]....
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Book•
01 Jan 2008
TL;DR: The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications.
Abstract: Mallat's book is the undisputed reference in this field - it is the only one that covers the essential material in such breadth and depth. - Laurent Demanet, Stanford University The new edition of this classic book gives all the major concepts, techniques and applications of sparse representation, reflecting the key role the subject plays in today's signal processing. The book clearly presents the standard representations with Fourier, wavelet and time-frequency transforms, and the construction of orthogonal bases with fast algorithms. The central concept of sparsity is explained and applied to signal compression, noise reduction, and inverse problems, while coverage is given to sparse representations in redundant dictionaries, super-resolution and compressive sensing applications. Features: * Balances presentation of the mathematics with applications to signal processing * Algorithms and numerical examples are implemented in WaveLab, a MATLAB toolbox * Companion website for instructors and selected solutions and code available for students New in this edition * Sparse signal representations in dictionaries * Compressive sensing, super-resolution and source separation * Geometric image processing with curvelets and bandlets * Wavelets for computer graphics with lifting on surfaces * Time-frequency audio processing and denoising * Image compression with JPEG-2000 * New and updated exercises A Wavelet Tour of Signal Processing: The Sparse Way, third edition, is an invaluable resource for researchers and R&D engineers wishing to apply the theory in fields such as image processing, video processing and compression, bio-sensing, medical imaging, machine vision and communications engineering. Stephane Mallat is Professor in Applied Mathematics at cole Polytechnique, Paris, France. From 1986 to 1996 he was a Professor at the Courant Institute of Mathematical Sciences at New York University, and between 2001 and 2007, he co-founded and became CEO of an image processing semiconductor company. Companion website: A Numerical Tour of Signal Processing * Includes all the latest developments since the book was published in 1999, including its application to JPEG 2000 and MPEG-4 * Algorithms and numerical examples are implemented in Wavelab, a MATLAB toolbox * Balances presentation of the mathematics with applications to signal processing
2,600 citations
"Empirical Wavelet Transform" refers background in this paper
...For further details, we refer the reader to the extensive literature about the wavelet theory, see for example [3], [10], [12], [14]....
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TL;DR: It turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions, and the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.
Abstract: Empirical mode decomposition (EMD) has recently been pioneered by Huang et al. for adaptively representing nonstationary signals as sums of zero-mean amplitude modulation frequency modulation components. In order to better understand the way EMD behaves in stochastic situations involving broadband noise, we report here on numerical experiments based on fractional Gaussian noise. In such a case, it turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions. It is also pointed out that the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.
2,304 citations
TL;DR: This paper introduces a precise mathematical definition for a class of functions that can be viewed as a superposition of a reasonably small number of approximately harmonic components, and proves that the method does indeed succeed in decomposing arbitrary functions in this class.
1,704 citations
"Empirical Wavelet Transform" refers background or methods in this paper
...In this paper, as we will use the Fourier formalism in Section III, we adopt the description used in [4] which is slightly different from the original used in [9]....
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...[4] entitled “synchrosqueezed wavelets”....
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...In [4], the authors proposed to model a mode as an amplitude modulated-frequency modulated (AM-FM) signal and then use the properties of such signals to build a functional to represent the whole signal....
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