Multidimensional empirical mode decomposition: Difference between revisions

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In [[signal processing]], '''multidimensional empirical mode decomposition''' ('''multidimensional EMD''') is an extension of the [[one-dimensional]] (1-D) [[Hilbert–Huang transform|EMD]] algorithm to a signal encompassing multiple-dimensional signaldimensions. The [[Hilbert–Huang transform|Hilbert–Huang empirical mode decomposition]] (EMD) process decomposes a signal into intrinsic mode functions combined with the [[Hilbert spectral analysis]], known as the [[Hilbert–Huang transform]] (HHT). The multidimensional EMD extends the 1-D [[Hilbert–Huang transform|EMD]] algorithm into multiple-dimensional signals. This decomposition can be applied to [[image processing]], [[audio signal processing]], and various other multidimensional signals.

==Motivation==

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The EMD method was developed so that data can be examined in an adaptive time–frequency–amplitude space for nonlinear and non-stationary signals.

The EMD method decomposes the input signal into fewseveral Intrinsic Mode Functions (IMF) and a residue. The given equation will be as follows:

: <math>I(n)=\sum_{m=1}^M \operatorname{IMF}_m(n)+\operatorname{Res}_M(n)</math>

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To improve the accuracy of measurements, the ensemble mean is a powerful approach, where data are collected by separate observations, each of which contains different noise over an ensemble of universe's. To generalize this ensemble idea, noise is introduced to the single data set, <math>x(t)</math>, as if separate observations were indeed being made as an analogue to a physical experiment that could be repeated many times. The added [[white noise]] is treated as the possible random noise that would be encountered in the measurement process. Under such conditions, the i th ‘artificial’ observation will be <math>x_i(t)=x(t)+w_i(t)</math>

In the case of only one observation, one of the multiple-observation ensembles is mimicked by adding not arbitrary but different copies of white noise, <math>w_i(t)</math>, to that single observation as given in the equation. Although adding noise may result in a smaller signal to-noise ratio, the added white noise will provide a uniform reference scale distribution to facilitate EMD; therefore, the low signal-noise ratio does not affect the decomposition method but actually enhances it by avoiding mode mixing. Based on this argument, an additional step is taken by arguing that adding white noise may help extract the true signals in the data, a method that is termed Ensemble Empirical Mode Decomposition (EEMD)

The EEMD consists of the following steps:

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=== Pseudo-bi-dimensional empirical mode decomposition<ref name=":5" /> ===

It should be pointed out here that the “pseudo-BEMD” method is not limited to only one-spatial dimension; rather, it can be applied to data of any number of spatial-temporal dimensions. Since the spatial structure is essentially determined by timescales of the variability of a physical quantity at each ___location and the decomposition is completely based on the characteristics of individual time series at each spatial ___location, there is no assumption of spatial coherent structures of this physical quantity. When a coherent spatial structure emerges, it better reflects the physical processes that drive the evolution of the physical quantity on the timescale of each component. Therefore, we expect this method to have significant applications in spatial-temporal data analysis.

To design a pseudo-BEMD algorithm the key step is to translate the algorithm of the 1D [[Hilbert huang transform|EMD]] into a Bi-dimensional Empirical Mode Decomposition (BEMD) and further extend the algorithm to three or more dimensions which is similar to the BEMD by extending the procedure on successive dimensions. For a 3D data cube of <math>i \times j \times k</math> elements, the pseudo-BEMD will yield detailed 3D components of <math>m \times n \times q</math> where <math>m</math>, <math>n</math> and <math>q</math> are the number of the IMFs decomposed from each dimension having <math>i</math>, <math>j</math>, and <math>k</math> elements, respectively.