Multidimensional empirical mode decomposition: Difference between revisions

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In [[signal processing]], the '''multidimensional empirical mode decomposition''' ('''multidimensional EMD''') is thean extension of the 1-D [[Hilbert–Huang transform|EMD]] algorithm intoto a multiple-dimensional signal. 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 [[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 "empirical mode decomposition" method can extract global structure and deal with fractal-like signals.

The EMD method was developed so that the 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 few Intrinsic Mode functionsFunctions (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>

where <math>I(n)</math> is the multi-component signal. <math>\operatorname{IMF}_m(n)</math> is the <math>M^\text{th}</math> intrinsic mode function, and <math>\operatorname{Res}_M(n)</math> represents the residue corresponding to <math>M</math> intrinsic modes.

==Ensemble empirical mode decomposition==

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, wi<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)

method but actually enhances it to avoid the mode mixing. Based on this argument, an additional step is taken by arguing that adding white noise may help to 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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# Adding a white noise series to the original data.

# Decomposing the data with added white noise into oscillatory components.

# Repeating step 1 and step 2 again and again, but with a different white noise series added each time.

# Obtaining the (ensemble) meansmean of the corresponding intrinsic mode functions of the decomposition as the final result.

In these steps, EEMD uses two properties of white noise:

# The added white noise leads to relatively even distribution of extrema distribution on all timescales.

# The [[Dyadic transformation|dyadic]] filter bank property provides a control on the periods of oscillations contained in an oscillatory component, significantly reducing the chance of scale mixing in a component. Through ensemble average, the added noise is averaged out.<ref name=":9" />

=== 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 reflects 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.