Multidimensional empirical mode decomposition

In signal processing, multidimensional empirical mode decomposition (multidimensional EMD) is an extension of the one-dimensional (1-D) EMD algorithm to a signal encompassing multiple dimensions. The 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 EMD algorithm into multiple-dimensional signals. This decomposition can be applied to image processing, audio signal processing, and various other multidimensional signals.

Motivation Multidimensional empirical mode decomposition is a popular method because of its applications in many fields, such as texture analysis, financial applications, image processing, ocean engineering, seismic research, etc. Several methods of Empirical Mode Decomposition have been used to analyze characterization of multidimensional signals.

Introduction to empirical mode decomposition (EMD)

The empirical mode decomposition (EMD) method can extract global structure and deal with fractal-like signals. 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 several intrinsic mode functions (IMF) and a residue. The given equation will be as follows:

I ( n ) = ∑ m = 1 M IMF m ⁡ ( n ) + Res M ⁡ ( n ) {\displaystyle I(n)=\sum _{m=1}^{M}\operatorname {IMF} _{m}(n)+\operatorname {Res} _{M}(n)}

where I ( n ) {\displaystyle I(n)} is the multi-component signal. IMF m ⁡ ( n ) {\displaystyle \operatorname {IMF} _{m}(n)} is the M th {\displaystyle M^{\text{th}}} intrinsic mode function, and Res M ⁡ ( n ) {\displaystyle \operatorname {Res} _{M}(n)} represents the residue corresponding to M {\displaystyle M} intrinsic modes.

Ensemble empirical mode decomposition The ensemble mean is an approach to improving the accuracy of measurements. Data is collected by separate observations, each of which contains different noise over an ensemble of universes. To generalize this ensemble idea, noise is introduced to the single data set, x ( t ) {\displaystyle x(t)} , 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 artificial ‘observation’ will be x i ( t ) = x ( t ) + w i ( t ) {\displaystyle x_{i}(t)=x(t)+w_{i}(t)} . In the case of only one observation, one of the multiple-observation ensembles is mimicked by adding different copies of white noise, w i ( t ) {\displaystyle w_{i}(t)} , 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 will 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:

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 mean 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 a relatively even distribution of extrema distribution on all timescales. The 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.

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