**3.3 Bidimensional EEMD (BEEMD)**

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While the value of *i* increases, lower frequency components are obtained. An example is given in Fig. 5a, where the signal is decomposed into five IMFs plus residue. The process to calculate each ܿ(t) is called sifting process. The local extrema are defined and interpolated, resulting in two fitting curves, one for the maxima and one for the minima. Then, the mean curve is calculated and subtracted from the signal. This procedure continues until a stopping criterion is satisfied. The signal that remains after the last subtraction is ܿଵ(t). Next, ܿଵ(t) is subtracted from the initial signal and the remainder constitutes the new initial signal on which the above procedure is applied in order to extract the following IMFs until the

Despite the great advantage of EMD, deficiency arises when the extrema of the original signal are unevenly distributed. In such a case, the IMFs are incorrectly calculated, since either a single IMF contains signals of widely disparate scales or a single mode of oscillations resides in two or more IMFs. This phenomenon is called *mode mixing* and an example is depicted in Fig. 5b. It is clear that the first two IMFs, apart from the high frequency component of the signal, incorrectly include a low frequency oscillation. To overcome this issue, Huang *et al.* proposed a noise-assisted version of EMD, namely ensemble EMD (EEMD) (Wu & Huang, 2009). EEMD requires the generation of an ensemble that contains multiple copies of the original signal that are distorted by white Gaussian noise, different for each copy, of finite amplitude. EMD is applied on every member of the ensemble and the final IMFs of the initial signal are derived by averaging the corresponding IMFs of each member of the ensemble. The concept of EEMD is grounded on the intuitive characteristics of white noise. White noise populates the whole time-frequency space uniformly and, as a result, establishes proper reference scales for the IMFs. The inherent modes of the signal are triggered by the noise and are projected accurately on the correct

Fig. 5. (a) EMD analysis, (b) mode mixing phenomenon, (c) ensemble EMD analysis.

desired number is obtained.

**3.2 Ensemble EMD (EEMD)** 

A multidimensional approach of EMD is required in case of a multidimensional signal. The extension of EMD in two dimensions (2D), namely Bidimensional EMD (BEMD), is an alternative multi-resolution analysis technique for image analysis and pattern discrimination. BEMD decomposes a 2D signal in 2D IMFs in the same way as eq. (1) demonstrates. However, there are two approaches for the realization of 2D extension. The first approach treats 2D data (images) as a collection of 1D slices (rows/columns) and applies 1D EMD on each row/column of the image (pseudo-BEMD). The second approach directly transplants the idea of 1D EMD algorithm in 2D data (genuine BEMD) after applying the appropriate changes (for example, fitting surfaces replace fitting curves). The first approach has the advantage of higher speed, while the latter exhibits improved performance, since the correlation among rows/columns of the image is taken into account. Bidimensional EEMD is the extension of EEMD in 2D (Wu *et al.*, 2009).
