There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

[1] Data analysis has been one of the core activities in scientific research, but limited by the availability of analysis methods in the past, data analysis was often relegated to data processing. To accommodate the variety of data generated by nonlinear and nonstationary processes in nature, the analysis method would have to be adaptive. Hilbert-Huang transform, consisting of empirical mode decomposition and Hilbert spectral analysis, is a newly developed adaptive data analysis method, which has been used extensively in geophysical research. In this review, we will briefly introduce the method, list some recent developments, demonstrate the usefulness of the method, summarize some applications in various geophysical research areas, and finally, discuss the outstanding open problems. We hope this review will serve as an introduction of the method for those new to the concepts, as well as a summary of the present frontiers of its applications for experienced research scientists.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2007RG000228

A review on Hilbert‐Huang transform: Method and its applications to geophysical studies

A review on empirical mode decomposition in fault diagnosis of rotating machinery

Image analysis by bidimensional empirical mode decomposition

This paper presents an extensive review on the artifact removal algorithms used to remove the main sources of interference encountered in the electroencephalogram (EEG), specifically ocular, muscular and cardiac artifacts. We first introduce background knowledge on the characteristics of EEG activity, of the artifacts and of the EEG measurement model. Then, we present algorithms commonly employed in the literature and describe their key features. Lastly, principally on the basis of the results provided by various researchers, but also supported by our own experience, we compare the state-of-the-art methods in terms of reported performance, and provide guidelines on how to choose a suitable artifact removal algorithm for a given scenario. With this review we have concluded that, without prior knowledge of the recorded EEG signal or the contaminants, the safest approach is to correct the measured EEG using independent component analysis-to be precise, an algorithm based on second-order statistics such as second-order blind identification (SOBI). Other effective alternatives include extended information maximization (InfoMax) and an adaptive mixture of independent component analyzers (AMICA), based on higher order statistics. All of these algorithms have proved particularly effective with simulations and, more importantly, with data collected in controlled recording conditions. Moreover, whenever prior knowledge is available, then a constrained form of the chosen method should be used in order to incorporate such additional information. Finally, since which algorithm is the best performing is highly dependent on the type of the EEG signal, the artifacts and the signal to contaminant ratio, we believe that the optimal method for removing artifacts from the EEG consists in combining more than one algorithm to correct the signal using multiple processing stages, even though this is an option largely unexplored by researchers in the area.

https://iopscience.iop.org/article/10.1088/1741-2560/12/3/031001/pdf

EEG artifact removal?state-of-the-art and guidelines

The main contribution of our approach is to apply the Hilbert-Huang Transform (which consists of two parts: (a) Empirical Mode Decomposition (EMD), and (b) the Hilbert spectral analysis) to texture analysis. The EMD is locally adaptive and suitable for analysis of non-linear or non-stationary processes. This one-dimensional decomposition technique extracts a finite number of oscillatory components or "well-behaved" AM-FM functions, called Intrinsic Mode Function (IMF), directly from the data. Firstly, we extend the EMD to 2D-data (i.e. images), the so called bidimensional EMD (BEMD), the process being called 2D-sifting process. The 2D-sifting process is performed in two steps: extrema detection by neighboring window or morphological operators and surface interpolation by radial basis functions or multigrid B-splines. Secondly, we analyse each 2D-IMF obtained by BEMD by studying local properties (amplitude, phase, isotropy and orientation) extracted from the monogenic signal of each one of them. The monogenic signal is a 2D-generalization of the analytic signal, where the Riesz Transform replaces the Hilbert Transform. The performance of this texture analysis method, using the BEMD and Riesz Transform, is demonstrated with both synthetic and natural images.

Texture analysis based on local analysis of the Bidimensional Empirical Mode Decomposition

In this paper, we propose some recent works on data analysis and synthesis based on Empirical Mode Decomposition (EMD). Firstly, a direct 2D extension of original Huang EMD algorithm with application to texture analysis, and fractional Brownian motion synthesis. Secondly, an analytical version of EMD based on PDE in 1D-space is presented. We proposed an extension in 2D-case of the so-called "sifting process" used in the original Huang's EMD. The 2D-sifting process is performed in two steps: extrema detection (by neighboring window or morphological operators) and surface interpolation by splines (thin plate splines or multigrid B-splines). We propose a multiscale segmentation approach by using the zero-crossings from each 2D-intrinsic mode function (IMF) obtained by 2D-EMD. We apply the Hilbert–Huang transform (which consists of two parts: (a) Empirical mode decomposition, and (b) the Hilbert spectral analysis) to texture analysis. We analyze each 2D-IMF obtained by 2D-EMD by studying local properties (amplitude, phase, isotropy, and orientation) extracted from the monogenic signal of each one of them. The monogenic signal proposed by Felsberg et al. is a 2D-generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. These local properties are obtained by the structure multivector such as proposed by Felsberg and Sommer. We present numerical simulations of fractional Brownian textures. Recent works published by Flandrin et al. relate that, in the case of fractional Gaussian noise (fGn), EMD acts essentially as a dyadic filter bank that can be compared to wavelet decompositions. Moreover, in the context of fGn identification, Flandrin et al. show that variance progression across IMFs is related to Hurst exponent H through a scaling law. Starting with these results, we proposed an algorithm to generate fGn, and fractional Brownian motion (fBm) of Hurst exponent H from IMFs obtained from EMD of a White noise, i.e., ordinary Gaussian noise (fGn with H = 1/2). Delechelle et al. proposed an analytical approach (formulated as a partial differential equation (PDE)) for sifting process. This PDE-based approach is applied on signals. The analytical approach has a behavior similar to that of the EMD proposed by Huang.

/pdf/empirical-mode-decomposition-applications-on-signal-and-4zp8gedwhc.pdf

Empirical mode decomposition: applications on signal and image processing

https://www.hal.inserm.fr/inserm-00588301/document

Multivariate empirical mode decomposition and application to multichannel filtering

We present a texture analysis algorithm based on gray-level cooccurrence (GLC) model and bidimensional empirical mode decomposition (BEMD) of a texture field. The EMD, which has been recently introduced in signal processing by Huang in 1998, is adaptive for nonlinear and nonstationary data analysis. The main contribution of our approach is to apply the empirical mode decomposition to texture extraction and image denoising. This decomposition, obtained by the bidimensional sifting process, plays an important role in the characterization of regions in textured images. The sifting process is realized using morphological operators to detect regional extrema and thanks to radial basis functions (RBF) for interpolation. We modified the original sifting process to permit a texture decomposition of images by inserting criteria proposed by second-order statistics from GLCs.

Jean-Claude Nunes

Papers

Image analysis by bidimensional empirical mode decomposition

Texture analysis based on local analysis of the Bidimensional Empirical Mode Decomposition

Empirical mode decomposition: applications on signal and image processing

Multivariate empirical mode decomposition and application to multichannel filtering

Texture analysis based on the bidimensional empirical mode decomposition with gray-level co-occurrence models