B-SPLINE ANALYTICAL REPRESENTATION OF THE MEAN ENVELOPE FOR EMPIRICAL MODE DECOMPOSITION

Author(s):  
TIANXIANG ZHENG ◽  
LIHUA YANG

This paper investigates how the mean envelope, the subtrahend in the sifting procedure for the Empirical Mode Decomposition (EMD) algorithm, represents as an expansion in terms of basis. To this end, a novel approach that gives an alternative analytical expression using B-spline functions is presented. The basic concept lies mainly on the idea that B-spline functions form a basis for the space of splines and have refined-node representations by knot insertion. This newly-developed expression is essentially equivalent to the conventional one, but gives a more explicit formulation on this issue. For the purpose of establishing the mathematical foundation of the EMD methodology, this study may afford a favorable opportunity in this direction.

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Hongyi Li ◽  
Chaojie Wang ◽  
Di Zhao

A B-spline empirical mode decomposition (BEMD) method is proposed to improve the celebrated empirical mode decomposition (EMD) method. The improvement of BEMD on EMD mainly concentrates on the sifting process. First, instead of the curve that resulted from computing the average of upper and lower envelopes, the curve interpolated by the midpoints of local maximal and minimal points is used as the mean curve, which can reduce the cost of computation. Second, the cubic spline interpolation is replaced with cubic B-spline interpolation on account of the advantages of B-spline over polynomial spline. The effectiveness of BEMD compared with EMD is validated by numerical simulations and an application to find the basis functions of EMI signals.


2019 ◽  
Vol 73 (12) ◽  
pp. 1436-1450 ◽  
Author(s):  
Fabiola León-Bejarano ◽  
Martin O. Méndez ◽  
Miguel G. Ramírez-Elías ◽  
Alfonso Alba

A novel method based on the Vancouver Raman algorithm (VRA) and empirical mode decomposition (EMD) for denoising Raman spectra of biological samples is presented. The VRA is one of the most used methods for denoising Raman spectroscopy and is composed of two main steps: signal filtering and polynomial fitting. However, the signal filtering step consists in a simple mean filter that could eliminate spectrum peaks with small intensities or merge relatively close spectrum peaks into one single peak. Thus, the result is often sensitive to the order of the mean filter, so the user must choose it carefully to obtain the expected result; this introduces subjectivity in the process. To overcome these disadvantages, we propose a new algorithm, namely the modified-VRA (mVRA) with the following improvements: (1) to replace the mean filter step by EMD as an adaptive parameter-free signal processing method; and (2) to automate the selection of polynomial degree. The denoising capabilities of VRA, EMD, and mVRA were compared in Raman spectra of artificial data based on Teflon material, synthetic material obtained from vitamin E and paracetamol, and biological material of human nails and mouse brain. The correlation coefficient (ρ) was used to compare the performance of the methods. For the artificial Raman spectra, the denoised signal obtained by mVRA ([Formula: see text]) outperforms VRA ([Formula: see text]) for moderate to high noise levels whereas mVRA outperformed EMD ([Formula: see text]) for high noise levels. On the other hand, when it comes to modeling the underlying fluorescence signal of the samples (i.e., the baseline trend), the proposed method mVRA showed consistent results ([Formula: see text]. For Raman spectra of synthetic material, good performance of the three methods ([Formula: see text] for VRA, [Formula: see text] for EMD, and [Formula: see text] for mVRA) was obtained. Finally, in the biological material, mVRA and VRA showed similar results ([Formula: see text] for VRA, [Formula: see text] for EMD, and [Formula: see text] for mVRA); however, mVRA retains valuable information corresponding to relevant Raman peaks with small amplitude. Thus, the application of EMD as a filter in the VRA method provides a good alternative for denoising biological Raman spectra, since the information of the Raman peaks is conserved and parameter tuning is not required. Simultaneously, EMD allows the baseline correction to be automated.


2010 ◽  
Vol 159 ◽  
pp. 377-382
Author(s):  
Guang Tao Ge

Define the course of getting mean envelope as an operation (mean envelope operation) in Empirical mode decomposition (EMD), so as to express the Intrinsic Mode Function (IMF) with mean envelopes. Summarize several rules of the mean envelope operation. On this fundamental, the abnormal components exist in the over-sifting IMFs are extracted out, and the conclusion is testified with the infinite sifting experiment.


Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. V403-V413 ◽  
Author(s):  
Julián L. Gómez ◽  
Danilo R. Velis

We developed a new and simple method for denoising seismic data, which was inspired by data-driven empirical mode decomposition (EMD) algorithms. The method, which can be applied either as a trace-by-trace process or in the [Formula: see text] domain, replaces the use of the cubic interpolation scheme, which is required to calculate the mean envelopes of the signal and the residues, by window averaging. The resulting strategy is not viewed as an EMD per se, but a user-friendly version of EMD-based algorithms that permits us to attain, in a fraction of the time, the same level of noise cancellation as standard EMD implementations. Furthermore, the proposed method requires less user intervention and easily processes millions of traces in minutes rather than in hours as required by conventional EMD-based techniques on a standard PC. We compared the performance of the new method against standard EMD methods in terms of computational cost and signal preservation and applied them to denoise synthetic and field (microseismic and poststack) data containing random, erratic, and coherent noise. The corresponding [Formula: see text] EMDs implementations for lateral continuity enhancement were analyzed and compared against the classical [Formula: see text] deconvolution to test the method.


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