Multidimensional empirical mode decomposition: Difference between revisions

Revision as of 04:45, 16 September 2022

Revision as of 13:07, 24 October 2022

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</math><ref name=":5" />

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 <math>X(i,j)=\sum_{ k \mathop =1}^mRX(k,i,j)</math>. <ref name=":5" />

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.

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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

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given as <math>I=f(x_1,x_2,x_3,x_4,\ldots,x_n)</math>

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 tilluntil 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

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The [[principal component analysis]]/[[Empirical orthogonal functions|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).

Typically, the EOFs are found by computing the eigenvalues and eigen vectors of a spatially weighted anomaly covariance matrix of a field. Most commonly, the spatial weights are the cos(latitude) or, better for EOF analysis, the sqrt(cos(latitude)). The derived eigenvalues provide a measure of the percent variance explained by each mode. Unfortunately, the eigenvalues are not necessarily distinct due to sampling issues. North et al. (Mon. Wea. Rev., 1982, eqns 24-2624–26) provide a 'rule of thumb' for determining if a particular eigenvalue (mode) is distinct from its nearest neighbor.

Atmospheric and oceanographic processes are typically 'red' which means that most of the variance (power) is contained within the first few modes. The time series of each mode (aka, principle components) are determined by projecting the derived eigen vectors onto the spatially weighted anomalies. This will result in the amplitude of each mode over the period of record.

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=== Fast multidimensional ensemble empirical mode decomposition<ref name=":7" /> ===

For a temporal signal of length ''M'', the complexity of cubic spline sifting through its local extrema is about the order of ''M,'' and so is that of the EEMD as it only repeats the spline fitting operation with a number that is not dependent on ''M''. However, as the sifting number (often selected as 10) and the ensemble number (often a few hundred) multiply to the spline sifting operations, hence the EEMD is time -consuming compared with many other time series analysis methods such as Fourier transforms and wavelet transforms. The MEEMD employs EEMD decomposition of the time series at each division grids of the initial temporal signal, the EEMD operation is repeated by the number of total grid points of the ___domain. The idea of the fast MEEMD is very simple. As PCA/EOF-based compression expressed the original data in terms of pairs of PCs and EOFs, through decomposing PCs, instead of time series of each grid, and using the corresponding spatial structure depicted by the corresponding EOFs, the computational burden can be significantly reduced.

The fast MEEMD includes the following steps:

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===Concept===

There are some problems in BEMD and boundary extending implementation in the iterative sifting process, including time -consuming, shape and continuity of the edges, decomposition results comparison and so on. In order to fix these problems, the '''Boundary Processing in Bidimensional Empirical Decomposition (BPBEMD)''' method was created. The main points of the new method algorithm will be described next.

===BPBEMD algorithm<ref name=":3" />===

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===Advantages===

This method can process larger number of elements than traditional BEMD method. Also, it can shorten the time -consuming for the process. Depended on using nonparametric sampling based texture synthesis, the BPBEMD could obtain better result after decomposing and extracting.

===Limitations===

Because most of image inputs are non-stationary which don’tdon't exist boundary problems, the BPBEMD method is still lack of enough evidence that it is adaptive to all kinds of input data. Also, this method is narrowly restricted to be use on texture analysis and image processing.

==Applications==

In the first part, these MEEMD techniques can be used on Geophysical data sets such as climate, magnetic, seismic data variability which takes advantage of the fast algorithm of MEEMD. The MEEMD is often used for nonlinear geophysical data filtering due to its fast algorithms and its ability to handle large amount of data sets with the use of compression without losing key information. The IMF can also be used as a signal enhancement of Ground Penetrating Radar for nonlinear data processing; it is very effective to detect geological boundaries from the analysis of field anomalies.<ref name=":6" />

In the second part, the PDE-based MEMD and FAMEMD can be implemented on audio processing, image processing and texture analysis. Because of its several properties, including stability, less time -consuming and so on, PDE-based MEMD method works well for adaptive decomposition, data denoising and texture analysis. Furthermore, the FAMEMD is a great method to reduce computation time and have a precise estimation in the process. Finally, the BPBEMD method has good performance for image processing and texture analysis due to its property to solve the extension boundary problems in recent techniques.

==References==