Multidimensional empirical mode decomposition
The empirical mode decomposition (EMD) method can extract global structure and deal with fractal-like signals.
The EMD method decomposes the input signal into several intrinsic mode functions (IMF) and a residue.
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 key part of the method is in the construction of the IMF according to the principle of combination of the comparable minimal scale components.
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 first step of performing sifting is to determine the upper and lower envelopes encompassing all the data by using the spline method.
Hence, it could exceed the computation capacity for a Geo-Physical data processing system when the number of EMD in the algorithm is large.
Typically, the EOFs are found by computing the eigenvalues and eigen vectors of a spatially weighted anomaly covariance matrix of a field.
The time series of each mode (aka, principle components) are determined by projecting the derived eigen vectors onto the spatially weighted anomalies.
Two factors inhibit physical interpretation of EOFs: (i) The orthogonality constraint and (ii) the derived patterns may be domain dependent.
If the data subjected to PCA/EOF analysis is all white noise, all eigenvalues are theoretically equal and there is no preferred vector direction for the principal component in PCA/EOF space.
For the new approach of using spatial division, the total number of values in PCA/EOF representation is where Therefore, the compression rate of the spatial domain is as follows The advantage of this algorithm is that an optimized division and an optimized selection of PC/EOF pairs for each region would lead to a higher rate of compression and result into significantly lower computation as compared to a Pseudo BEMD extended to higher dimensions.
Source:[4] 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 fast MEEMD includes the following steps: In this compressed computation, we have used the approximate dyadic filter bank properties of EMD/EEMD.
[5] The EEMDs comprising MEEMD are assigned to independent threads for parallel execution, relying on the OpenMP runtime to resolve any load imbalance issues.
The impact of the unavoidable branch divergence from data irregularity, caused by the noise, is minimized via a regularization technique using the on-chip memory.
[5] The fast and adaptive bidimensional empirical mode decomposition (FABEMD) is an improved version of traditional BEMD.
Therefore, the MAX and MIN filter will form a new 2-D matrix for envelope surface which will not change the original 2-D input data.
This method (FABEMD) provides a way to use less computation to obtain the result rapidly, and it allows us to ensure more accurate estimation of the BIMFs.
The Partial Differential Equation-Based Multidimensional Empirical Mode Decomposition (PDE-based MEMD) approach is a way to improve and overcome the difficulties of mean-envelope estimation of a signal from the traditional EMD.
In order to perform multidimensional EMD, we need to extend the 1-D PDE-based sifting process[9] to 2-D space as shown by the steps below.
Secondly, we can use additive operator splitting (AOS)[10] scheme to improve the property of stability, because the small time step
Based on the Navier–Stokes equations directly, this approach provides a good way to obtain and develop theoretical and numerical results.
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.
Depended on using nonparametric sampling based texture synthesis, the BPBEMD could obtain better result after decomposing and extracting.
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.
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.
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.
