Multivariate empirical mode decomposition and application to multichannel filtering

The vibration measured from wind turbine drivetrain components is a mixture of multiple frequency modes. In practice, in wind turbine drivetrain condition monitoring systems, multiple accelerometer sensors are used to measure the vibration. Inter-channel common modes are not processed in the standard single-channel empirical mode decomposition (EMD) and it suffers from mode mixing and mode misalignment. Inter-channel correlation implies the causation of vibration mode shapes. Multivariate EMD (MEMD) possesses an enhanced spatial and spectral coherence. The mode alignment property of MEMD is used to process the inter-channel common modes, thus MEMD overcomes the limitation of mode misalignment in single-channel EMD. Still, MEMD exhibits a degree of mode mixing. White noise powers are added in separate channels to lessen the mode mixing. In this research, a novel multivariate signal processing technique, noise-assisted multivariate empirical mode signal decomposition (NA-MEMD) with a competent nonlinear Teager-Kaiser energy operator (NLTKEO), is proposed and tested for truthful extraction of instantaneous frequency and instantaneous amplitude features, and thereby ensures superior fault diagnosis performance. The dyadic filter bank structure of the proposed NA-MEMD decomposes the nonstationary vibrations effectively. The proposed method is used to predict the surface damage pattern embedded in multi-source vibrations at a low-speed planetary gear stage. The effectiveness of the proposed algorithm is tested with NREL GRC wind turbine condition monitoring benchmark datasets.

We propose a novel class of schemes for the pansharpening of multispectral (MS) images using a multivariate empirical mode decomposition (MEMD) algorithm. MEMD is an extension of the empirical mode decomposition (EMD) algorithm, which enables the decomposition of multivariate data into its intrinsic oscillatory scales. The ability of MEMD to process multichannel data directly by performing data-driven, local, and multiscale analysis makes it a perfect match for pansharpening applications, a task for which standard univariate EMD is ill-equipped due to the nonuniqueness, mode-mixing, and mode-misalignment issues. We show that MEMD overcomes the limitations of standard EMD and yields improved spatial and spectral performance in the context of pansharpening of MS images. The potential of the proposed schemes is further demonstrated through comparative analysis against a number of standard pansharpening algorithms on both simulated Pleiades and real-world IKONOS data sets.

A novel quadrature clutter rejection approach based on the multivariate empirical mode decomposition for bidirectional Doppler ultrasound signals

A novel scheme to perform the fusion of multiple images using the multivariate empirical mode decomposition (MEMD) algorithm is proposed. Standard multi-scale fusion techniques make a priori assumptions regarding input data, whereas standard univariate empirical mode decomposition (EMD)-based fusion techniques suffer from inherent mode mixing and mode misalignment issues, characterized respectively by either a single intrinsic mode function (IMF) containing multiple scales or the same indexed IMFs corresponding to multiple input images carrying different frequency information. We show that MEMD overcomes these problems by being fully data adaptive and by aligning common frequency scales from multiple channels, thus enabling their comparison at a pixel level and subsequent fusion at multiple data scales. We then demonstrate the potential of the proposed scheme on a large dataset of real-world multi-exposure and multi-focus images and compare the results against those obtained from standard fusion algorithms, including the principal component analysis (PCA), discrete wavelet transform (DWT) and non-subsampled contourlet transform (NCT). A variety of image fusion quality measures are employed for the objective evaluation of the proposed method. We also report the results of a hypothesis testing approach on our large image dataset to identify statistically-significant performance differences.

Large-eddy simulations of an SD7003 airfoil are conducted for an imposed plunging motion in order to induce a deep dynamic stall regime. Free stream flow parameters are set to Reynolds number Re = 60.000 and Mach number M = 0.1. Coherent structures are identified under the light of modal analysis via the multivariate empirical mode decomposition (EMD) and the multi resolution dynamic mode decomposition (mrDMD). The present study shows that the wavelet approach in the mrDMD does not fully compensate the errors in the frequency analysis related to rotational and translational invariances associated with vortex transport. Meanwhile the Hilbert-Huang transform applied to the intrinsic mode functions from EMD provides a more convenient way to identify the frequencies related to the coherent structures with negligible reconstruction error.

Currently, empirical mode decomposition (EMD) has become a popular data-driven time-frequency analysis method for nonstationary and nonlinear data. However, it is still limited to univariate data due to the number and/or scale misalignment for multivariate data. A newly developed multivariate EMD (MEMD) scheme decomposes multivariate data simultaneously and thus leads to mode alignment and minimizes mode mixing. In addition, an improved amplitude and frequency modulation (AM-FM) decomposition algorithm presented here provides an estimation of a more meaningful instantaneous amplitude and frequency than the widely used Hilbert transform (HT). Both of these facilitate development of a time-frequency analysis framework for multivariate nonstationary and nonlinear data analysis. In this paper, MEMD-based scalogram and coscalogram, and instantaneous frequency spectra and cospectra are proposed to characterize a multivariate nonstationary process. The scalogram and instantaneous frequency spectra capture spectral evolution of each component while the coscalogram and instantaneous frequency cospectra reveal embedded intermittent correlation between two components. Compared with scale-based scalogram and coscalogram, frequency-based instantaneous frequency spectra and cospectra provide a more detailed portrait of multivariate data. The effectiveness of the proposed MEMD-based time-frequency analysis framework is demonstrated by numerical examples of a thunderstorm downburst and an earthquake ground motion. Also, the results from the MEMD-based approach are compared with those based on a continuous wavelet transform, which further reinforces the effectiveness of the proposed framework.

Motion capture data are widely used in different research fields such as medical, entertainment, and industry. However, most motion researches using motion capture data are carried out in the time-domain. To understand human motion complexities, it is necessary to analyze motion data in the frequency-domain. In this paper, to analyze human motions, we present a framework to transform motions into the instantaneous frequency-domain using the Hilbert-Huang transform (HHT). The empirical mode decomposition (EMD) that is a part of HHT decomposes nonstationary and nonlinear signals captured from the real-world experiments into pseudo monochromatic signals, so-called intrinsic mode function (IMF). Our research reveals that the multivariate EMD can decompose complicated human motions into a finite number of nonlinear modes (IMFs) corresponding to distinct motion primitives. Analyzing these decomposed motions in Hilbert spectrum, motion characteristics can be extracted and visualized in instantaneous frequency-domain. For example, we apply our framework to (1) a jump motion, (2) a foot-injured gait, and (3) a golf swing motion.

The soft fault signal of non-linear analog circuits is non-linear and unstable. Its inherent characteristics cannot be analyzed by the traditional linear method. Therefore, a new method based on the empirical mode decomposition (EMD) and a generalized multi-fractal dimension algorithm is proposed for soft fault signal feature extraction of non-linear analog circuits. The EMD decomposed the circuits soft fault signal into several intrinsic mode functions (IMF). The EMD then modified a traditional G-P correlation dimension algorithm to simply and reliably use it for calculating the generalized multi-fractal dimensions of the fault signals. According to fractal theory, the generalized fractal dimensions of the original fault signals and IMF can be obtained for an appropriate weight factor q. The principle IMF component can then be filtered using the cross-correlation coefficient criterion. The obtained generalized multi-fractal dimension of the principle IMF component was continually input in the supporting vector machine as a characteristic vector to diagnose the soft fault of the non-linear analog circuits. The feasibility and validity of this proposed method were demonstrated by experiments.

Using filter bank property to simplify the calculations of Empirical Mode Decomposition

Envelope modified versions of the empirical mode decomposition (EMD) method such as the B‐spline interpolation‐based EMD (B‐EMD) method and cardinal spline interpolation‐based EMD (C‐EMD) method have been proposed recently for purpose of improving its effectiveness. To shed further light on their performance, the behaviours of these EMD‐type methods in the presence of white Gaussian noises are investigated in this study based on extensive numerical experiments. Similarly to the EMD method, it turns out that the envelope modified EMD methods also act as filter banks essentially. However, the spectra among the first several intrinsic mode functions of the B‐EMD method have fewer overlaps than those of the EMD and C‐EMD methods, which indicate that the B‐EMD method has a better ability to alleviate the mode mixing problem for signals with higher frequencies. On the other hand, the C‐EMD method is shown to perform better than the EMD and B‐EMD methods on separating tones with lower frequencies.

In recent years, the problem of heavy metal pollution in agriculture caused by industrial development has been particularly prominent, directly affecting food and ecological environmental safety. Hyperspectral remote sensing technology has the advantages of high spectral resolution and nondestructive monitoring. The physiological and biochemical parameters of crops undergo similar changes under different heavy metal stresses. Therefore, it is a great challenge to explore the use of hyperspectral technology to distinguish the types of the heavy metal copper (Cu) and lead (Pb) elements. This is also a hot topic in the current research. In this study, several models are proposed to distinguish copper and lead elements by combining multivariate empirical mode decomposition (MEMD) transformation and machine learning. First, MEMD is introduced to decompose the original spectrum, which effectively removes the noise and highlights and magnifies the weak information of the spectrum. The successive projections algorithm (SPA), competitive adaptive reweighted sampling (CARS), and iteratively retaining informative variables (IRIV) were used to screen the characteristic bands and were combined with extreme learning machine (ELM), support vector machine (SVM), and general regression neural network (GRNN) algorithms to build models to distinguish the types of Cu and Pb elements. The quality of the model was evaluated using accuracy (A), precision (P), recall (R), and F-score. The results showed that the MEMD-SPA-SVM, MEMD-CARS-SVM, MEMD-SPA-ELM, MEMD-CARS-ELM, and MEMD-IRIV-ELM models intuitively and effectively distinguished the types of Cu and Pb elements. Their accuracy and F-scores were all greater than 0.8. To verify the superiority of these models, the same model was constructed based on first derivative (FD) and second derivative (SD) transformations, and the obtained classification and recognition accuracy (A) and F-score were both lower than 0.8, which further confirmed the superiority of the model established after MEMD transformation. The model proposed in this study has great potential for applying hyperspectral technology to distinguish the types of elements contaminated by Cu and Pb in crops.

A synthetic aperture radar (SAR) target recognition method was proposed based on multivariate empirical mode decomposition (MEMD). MEMD was the general extension of traditional EMD, which could avoid the mode mixing problems. MEMD was employed to process SAR images to obtain the multi-layer intrinsic mode functions (IMF), which could better reflect the time-frequency properties of the targets. Different layers of IMFs could effectively complement each other while sharing some inner correlations because they are generated from the same target. In the classification phase, the joint sparse representation was employed to represent the IMFs. The joint sparse representation could solve several related sparse representation tasks based on the idea of multi-task learning. It could produce more precise estimations than the solutions of single tasks. According to the sparse coefficient vectors corresponding to different IMFs, the reconstruction errors of different classes for the representation of the test sample can be calculated. Afterwards, the target label of the test sample can be determined. Experiments were conducted on the Moving and Stationary Target Acquisition and Recognition (MSTAR) dataset, by comparison with existing methods under the standard operating condition, depression angle variance, noise corruption, and target occlusion, the results confirm the validity of the proposed method.

A novel time domain method based on improved multivariate empirical mode decomposition (MEMD) for plant-wide oscillations characterization is proposed. The original MEMD is ameliorated in the following aspects, (i) decorrelation of the two-dimensional Halton sequences, (ii) boundary processing to restrain end effect and (iii) improved criterion for sifting process stoppage. Due to its capability to analyze multiple channels data, MEMD is especially suitable for characterizing plant-wide control loop oscillations. Simulation examples and industrial case study are provided to demonstrate the effectiveness of the proposed MEMD based approach.

The non-stationary character of the acoustic signals and the overlapping issue of blockage features can be challenging for condition detection of the underground water supply pipes. Aiming to solve the above problems, a method including feature extraction and pattern recognition based on multivariate empirical mode decomposition (MEMD) and approximate entropy was proposed in this paper. Firstly, the multichannel acoustic signals which were collected from water supply pipes under working conditions were decomposed by using MEMD, a series of intrinsic mode function (IMF) of each signal are obtained. Then, according to the criterions combining correlation coefficient and the variance contribution rate, the IMF components which comprise the main information of the pipe condition are selected; the approximate entropy of the selected IMF components are calculated to construct the feature vectors. Finally, the support vector machine (SVM) is used to classify and identify the pipe conditions. Experimental results have shown that the proposed method is superior to the EMD method in processing multichannel data, and it is more accurate in detecting the degree of blocking.

The conventional microphone array near-field Fourier acoustic holography using Discrete Fourier Transform (DFT) is able to efficiently reconstruct sound field and acquire an image of noise distribution. However, Fourier transform causes measuring error in practical applications, and people have to select primary frequency for observing sound field holography based on the spectrum of source signal. In this paper, we use the empirical mode decomposition (EMD) owing to its completeness, orthogonality, and adaptiveness, which are able to decompose multiple sound sources in the spatial domain and acquire instantaneous frequencies via intrinsic mode functions (IMFs). Prior information about the primary frequency is not necessary by this approach that makes the simultaneous observation of each source possible. In addition, EMD sound source imaging approach may be integrated into a near-field equivalent source imaging (NESI) system, which includes a virtual microphone technology generally used for sound field image enhancement. We have implemented and compared the constituent 1D EMD, 2D EMD spatial transform systems, and EMD based NESI approach in Labview language. Several experimental results and detailed discussions are also provided to verify the characteristics of multiple sound sources.