Copyright © 2008 Nii Attoh-Okine et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Data from natural phenomena are usually
nonstationary due to their transient nature; also, the span of captured data
may be shorter than the longest time scale that describes the phenomenon. In
fact, since it is impossible or impractical to obtain infinite data points
describing a phenomenon, all data are invariably short. To simplify processing
and analysis, data stationarity is often assumed even though the condition may
not be strictly satisfied. For instance, the stationarity assumption justifies
traditional Fourier-based methods, which utilize a priori basis sets to
globally decompose a signal. To directly address the processing of
nonstationary and nonlinear signals, the Hilbert-Huang transform (HHT) has
recently been developed. The HHT comprises two steps: empirical mode decomposition
(EMD) and Hilbert spectral analysis (HSA). Unlike Fourier-based methods, the EMD
decomposes a signal into its components adaptively without using a priori
basis. The decomposition is based on the local time scale of the data. The
adaptive nature of the process successfully decomposes nonlinear, nonstationary
signals in the time domain. Moreover, the decomposition components, referred to
as intrinsic mode functions (IMFs), are generally in good agreement with
intuitive and physical signal interpretations. Moreover, the IMFs have
well-defined instantaneous frequencies. Accordingly, the HSA Hilbert transforms
the IMFs to generate a full energy-frequency-time plot (Hilbert spectrum),
which gives the instantaneous energy and frequency content of the signal. The
bidimensional empirical mode decomposition (BEMD) has recently been introduced
as a 2D extension to the EMD. Thus, the EMD and BEMD are increasingly being
employed to successfully address many contemporary signal processing applications.
Bidimensional empirical mode decomposition (BEMD) is an extension of the one-dimensional EMD
applied to two-dimensional signals. Images are usually decomposed with BEMD
using different interpolation methods to extract IMFs. An important aspect of
the BEMD is the construction of envelopes when sifting for IMFs, which involves
interpolation of scattered data formed by the extrema of the data. Three broad
methods of scattered data interpolation are radial basis function methods, triangulation-based
methods, and inverse distance weighted methods. In using any of these major
methods, there are two approaches to data interpolation: global and local
approaches. In the global approach, interpolated data are influenced by all
data within the given domain, whereas in the local approach, interpolated
values are influenced by data within a neighborhood of the interpolated points.
Global methods tend to be computationally costlier than local methods due to
the generation of larger coefficient matrices that can easily become highly ill-conditioned.
A number of issues have come up concerning empirical mode decomposition,
including the following.
(1)Finding mathematical and physical meaning for IMFs, since EMD is essentially algorithmic in nature and lacks mathematical
rigor.(2)Determining the most appropriate interpolation scheme.(3)Identifying criteria for stopping the sifting process.(4)Handling of boundary or end effects during data interpolation.Most success in EMD has been in 1D, however, one issue still persists in all these
advancements: the physical significance of IMFs derived from the original data
series or signal. A thorough understanding of the physical processes that
generate data is required before any form of scientific explanation can be
attributed to any particular IMF or group of IMFs. Even with this kind of
thorough knowledge, there is still a level of ambiguity when trying to extract
information from the IMFs that is directly relevant to the original signal and
the physics of the underlying system. Before getting to the point where
essential information can be extracted from the IMFs, there is a need to
determine which IMFs are really relevant to the decomposition process and which
carry the necessary information required to understand the underlying system, as
EMD is a numerical procedure with possible numerical errors in the results.
BEMD has potential in image preprocessing in
the area of edge detection. The first few IMFs in BEMD contain the highest
spatial frequencies contained in the original image, so that separating out
these first few IMFs can smooth out the image for further processing.
The purpose of this special
issue is to address the following issues in both 1D and 2D empirical mode
decompositions:
1.theoretical analysis and understanding; 2.performance enhancements of the EMD; 3.single decomposition, monitoring, and analysis; 4.feature extraction; 5.fast and adaptive methods; 6.decomposition domain processing methods; 7.image analysis and segmentation; 8.texture representation and segmentation; 9.optimization; 10.signal fusion and interpolation; 11.signal processing applications in Engineering and Biomedical.
This special issue contains 12 papers. Of these there are 5 theoretical papers. The article by
J. F. Khan et al. introduced a novel contour-based method for detecting largely
affine invariant interest or feature points. The main contribution of the paper
is the selection of good discriminative feature points from relatively thinned
edges. Repeatability rate, which evaluates the geometric stability under
different transformation, was used as the performance criteria. L. Yan et al.
developed a filtering approach to address the mode mixing problem caused by
intermittency signal in EMD process. The authors first used wavelet denoising
and then applied the EMD procedure. The results show that this filtering
approach affectively avoids the mode mixing and retain useful information. S.
McLaughin and Y. Kopsinis used double iterative sifting and high interpolation
in the EMD procedure. It appears that this approach has the capability of
improving the performance. Binwei Weng and Kenneth Barner developed a method
for signal reconstruction. The proposed reconstruction algorithm gives the best
estimate of a given signal in the minimum mean square error sense. The
algorithm involves two steps: (a) formulation of linear weighting for the IMF,
(b) bidirectional weighting. S. M. A. Bhuiyan et al. proposed a multiple
hierarchical method for BEMD. In the approach, order statistics are used to get
the upper and lower envelopes, where the filter size is derived from the data.
Two papers develop a hybrid
approach between SVM, clustering, and EMD. N. Logothetis et al. initially used
EMD procedure, and unsurpervised K-means clustering the IMF and exploiting the SVM
on the extracted features. The authors tested their methodology on local field
potential in monkey cortex for decoding its bistable structure-from-motion
perception. Yu Yang et al. EMD is used as preprocessor for AR (autoregressive)
analysis; SVM is then used to classifier the output.
There were few papers on the applications.
H. Snoussi et al. performed a comparative analysis of EMD and complex empirical
mode decomposition and bivariate empirical mode decomposition. The two new methods
appear to be suitable to complex time series. The authors applied their methodology
to posture analysis. Yanhui Liu et al. used the EMD procedure to develop a new
technique—multimodal pressure
flow method (MMPF) for assessment of
cerebral autoregulation. The results obtained by the authors for the new methodology
are applicable in engineering and biomedical applications. A. Bouchikhi et al.
used the EMD in speech enhancement. The authors used two strategies: filtering
and thresholding. The authors demonstrated that their propose approach performs
better than wavelet applications. Olivier Adam used EMD as segmentation of
killer whales vocalizations; the results were very favorable compared to the
alternative methods. Finally, N. O. Attoh-Okine and A. Ayenu-Prah [1] used the BEMD
to evaluate pavement image crack detection and classification. The work appears
to have general application in structural health monitoring in civil
infrastructure applications.
We sincerely hope that the
diverse papers in this special issue will introduce various researchers,
engineers, and students to this new emerging field. Although the EMD is at its
infancy, the number of papers keeps increasing astronomically every year.
Finally, we hope that more mathematicians will address some of the “mathematical
and theoretical” limitations.
Nii Attoh-Okine
Kenneth Barner
Daniel Bentil
Ray Zhang
References
- A. Ayenu-Prah, Empirical mode decomposition and civil infrastructure systems, Ph.D. dissertation, University of Delaware, Newark, Del, USA, 2007.