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:
where is the multi-component signal. is the intrinsic mode function, and represents the residue corresponding to 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, , 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 .
In the case of only one observation, one of the multiple-observation ensembles is mimicked by adding different copies of white noise, , 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.
Pseudo-bi-dimensional empirical mode decomposition Source:
The “pseudo-BEMD” method is not limited to 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 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 elements, the pseudo-BEMD will yield detailed 3D components of where , and are the number of the IMFs decomposed from each dimension having , , and elements, respectively.
Mathematically let us represent a 2D signal in the form of matrix form with a finite number of elements.
At first we perform EMD in one direction of X(i, j), Row wise for instance, to decompose the data of each row into m components, then to collect the components of the same level of m from the result of each row decomposition to make a 2D decomposed signal at that level of m. Therefore, m set of 2D spatial data are obtained
where RX (1, i, j), RX (2, i, j), and RX (m, i, j) are the m sets of signal as stated (also here we use R to indicate row decomposing). The relation between these m 2D decomposed signals and the original signal is given as .
The first row of the matrix RX (m, i, j) is the mth EMD component decomposed from the first row of the matrix X (i, j). The second row of the matrix RX (m, i, j) is the mth EMD component decomposed from the second row of the matrix X (i, j), and so on.
Suppose that the previous decomposition is along the horizontal direction, the next step is to decompose each one of the previously row decomposed components RX(m, i, j), in the vertical direction into n components. This step will generate n components from each RX component.
For example, the component
RX(1,i,j) will be decomposed into CRX(1,1,i,j), CRX(1,2,i,j),...,CRX(1,n,i,j) RX(2,i,j) will be decomposed into CRX(2,1,i,j), CRX(2,2,i,j),..., CRX(2,n,i,j) RX(m,i,j) will be decomposed into CRX(m,1,i,j), CRX(m,2,i,j),..., CRX(m,n,i,j)
where C means column decomposing. Finally, the 2D decomposition will result into m× n matrices which are the 2D EMD components of the original data X(i,j). The matrix expression for the result of the 2D decomposition is
where each element in the matrix CRX is an i × j sub-matrix representing a 2D EMD decomposed component. We use the arguments (or suffixes) m and n to represent the component number of row decomposition and column decomposition, respectively rather than the subscripts indicating the row and the column of a matrix. Notice that the m and n indicate the number of components resulting from row (horizontal) decomposition and then column (vertical) decomposition, respectively.
By combining the components of the same scale or the comparable scales with minimal difference will yield a 2D feature with best physical significance. The components of the first row and the first column are approximately the same or comparable scale although their scales are increasing gradually along the row or column. Therefore, combining the components of the first row and the first column will obtain the first complete 2D component (C2D1). The subsequent process is to perform the same combination technique to the rest of the components, the contribution of the noises are distributed to the separate component according to their scales. As a result, the coherent structures of the components emerge, In this way, the pseudo-BEMD method can be applied to reveal the evolution of spatial structures of data.
Following the convention of 1D EMD, the last component of the complete 2D components is called residue.
The decomposition scheme proposed here could be extended to data of any dimensions such as data of a solid with different density or other measurable properties
given as
In which the subscription, n, indicated the number of dimensions. The procedure is identical as stated above: the decomposition starts with the first dimension, and proceeds to the second and third until all the dimensions are exhausted. The decomposition is still implemented by slices. This new approach is based on separating the original data into one-dimensional slices, then applying ensemble EMD to each one-dimensional slice. The key part of the method is in the construction of the IMF according to the principle of combination of the comparable minimal scale components.
For example, the matrix expression for the result of a 3D decomposition is TCRX(m,n,q,i,j,k) where T denotes the depth (or time) decomposition. Based on the comparable minimal scale combination principle as applied in the 2D case, the number of complete 3D components will be the smallest value of m, n, and q. The general equation for deriving 3D components is
where denotes the level of the C3D i.e.
The pseudo-BEMD method has several advantages. For instance, the sifting procedure of the pseudo-BEMD is a combination of one dimensional sifting. It employs 1D curve fitting in the sifting process of each dimension, and has no difficulty as encountered in the 2D EMD algorithms using surface fitting, which has the problem of determining the saddle point as a local maximum or minimum. Sifting is the process which separates the IMF and repeats the process until the residue is obtained. The first step of performing sifting is to determine the upper and lower envelopes encompassing all the data by using the spline method. Sifting scheme for pseudo-BEMD is like the 1D sifting where the local mean of the standard EMD is replaced by the mean of multivariate envelope curves.
The major disadvantage of this method is that although we could extend this algorithm to any dimensional data we only use it for Two dimension applications. Because the computation time of higher dimensional data would be proportional to the number of IMF's of the succeeding dimensions. Hence, it could exceed the computation capacity for a Geo-Physical data processing system when the number of EMD in the algorithm is large. Hence, we have mentioned below faster and better techniques to tackle this disadvantage.
Multi-dimensional ensemble empirical mode decomposition Source:
A Fast and efficient data analysis is very important for large sequences hence the MDEEMD focuses on two important things
Data compression which involves decomposing data into simpler forms. EEMD on the compressed data; this is the most challenging since on decomposing the compressed data there is a high probability to lose key information. A data compression method that uses principal component analysis (PCA)/empirical orthogonal function (EOF) analysis or principal oscillation pattern analysis is used to compress data.
Principal component analysis (PCA) or empirical orthogonal function analysis (EOF)
The principal component analysis/empirical orthogonal function analysis (PCA/EOF) has been widely used in data analysis and image compression, its main objective is to reduce a data set containing a large number of variables to a data set containing fewer variables, but that still represents a large fraction of the variability contained in the original data set. In climate studies, EOF analysis is often used to study possible spatial modes (i.e., patterns) of variability and how they change with time . In statistics, EOF analysis is known as principal component analysis (PCA).