Multidimensional empirical mode decomposition: Difference between revisions

Revision as of 17:53, 6 February 2023

Latest revision as of 14:16, 12 February 2025

(12 intermediate revisions by 4 users not shown)

Line 1:

{{multiple issues|

{{context|date=May 2021}}

{{Technical|date=August 2022}}

}}

In [[signal processing]], '''multidimensional empirical mode decomposition''' ('''multidimensional EMD''') is an extension of the [[one-dimensional]] (1-D) [[Hilbert–Huang transform|EMD]] algorithm to a signal encompassing multiple dimensions. 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 the [[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.

Line 21 ⟶ 17:

==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, <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 ‘artificial’artificial observation‘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 different copies of white noise, <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 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).

Line 37 ⟶ 33:

# 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 ===
Source:<ref name=":5" /> ===

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.

Line 109 ⟶ 107:

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:<ref name=":7" /> ===

A Fast and efficient data analysis is very important for large sequences hence the MDEEMD focuses on two important things

Line 146:

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.

=== Fast multidimensional ensemble empirical mode decomposition ===
Source:<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.

Line 162 ⟶ 164:

Note that a detailed knowledge of the intrinsic mode functions of a noise corrupted signal can help in estimating the significance of that mode. It is usually assumed that the first IMF captures most of the noise and hence from this IMF we could estimate the Noise level and estimate the noise corrupted signal eliminating the effects of noise approximately. This method is known as denoising and detrending. Another advantage of using the MEEMD is that the mode mixing is reduced significantly due to the function of the EEMD.<br />The denoising and detrending strategy can be used for image processing to enhance an image and similarly it could be applied to Audio Signals to remove corrupted data in speech. The MDEEMD could be used to break down images and audio signals into IMF and based on the knowledge of the IMF perform necessary operations. The decomposition of an image is very advantageous for radar-based application the decomposition of an image could reveal land mines etc.

== Parallel implementation of multi-dimensional ensemble empirical mode decomposition.<ref name=":8" /> ==

In MEEMD, although ample parallelism potentially exists in the ensemble dimensions and/or the non-operating dimensions, several challenges still face a high performance MEEMD implementation.<ref name=":8" />

[[File:Sample_BEMD_Simulation_results_for_a_noisy_signal.jpg|thumb|Bi-Dimensional EMD corrupted with Noise]]

Line 176 ⟶ 178:

# Its implementation is like the sequential one, which makes it more straightforward.

=== OpenMpOpenMP implementation.<ref name=":8" /> ===

The EEMDs comprising MEEMD are assigned to independent threads for parallel execution, relying on the OpenMP runtime to resolve any load imbalance issues. Stride memory accesses of high-dimensional data are eliminated by transposing these data to lower dimensions, resulting in better utilization of cache lines. The partial results of each EEMD are made thread-private for correct functionality. TheMemory requiredrequirements memory dependsdepend on the number of OpenMP threads and isare managed by OpenMP runtime.<ref name=":8" />

=== CUDA implementation.<ref name=":8" /> ===

In the GPU CUDA implementation, each EMD, is mapped to a thread. The memory layout, especially of high-dimensional data, is rearranged to meet memory coalescing requirements and fit into the 128-byte cache lines. The data is first loaded along the lowest dimension and then consumed along a higher dimension. This step is performed when the Gaussian noise is added to form the ensemble data. In the new memory layout, the ensemble dimension is added to the lowest dimension to reduce possible branch divergence. 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. Moreover, the cache memory is utilized to amortize unavoidable uncoalesced memory accesses.<ref name=":8" />

Line 189 ⟶ 191:

Based on the algorithm of BEMD, the implementation method of FABEMD is really similar to BEMD, but the FABEMD approach just changes the interpolation step into a direct envelope estimation method and restricts the number of iterations for every BIMF to one. As a result, two order statistics, including MAX and MIN, will be used for approximating the upper and lower envelope. The size of the filter will depend on the maxima and minima maps obtained from the input. The steps of the FABEMD algorithm are listed below.

===FABEMD algorithm===
Source:<ref name=":0" />===

;Step 1 – Determine and detect local maximum and minimum

Line 257 ⟶ 260:

Here, we take 2-D PDE-based EMD as an example.

===PDE-based BEMD algorithm===
Source:<ref name=":2" />===

;Step 1 – Extend super diffusion model from 1-D to 2-D

Line 316 ⟶ 320:

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" />===

The few core steps for BPBEMD algorithm are:<ref name=":3" />

;Step 1