ap-6-10.dvi


Acta Polytechnica Vol. 50 No. 6/2010

Ensemble Empirical Mode Decomposition: Image Data Analysis with
White-noise Reflection

M. Kopecký

Abstract

During the last decade, Zhaohua Wu and Norden E. Huang announced a new improvement of the original Empirical
Mode Decomposition method (EMD). Ensemble Empirical Mode Decomposition and its abbreviation EEMD represents
a major improvement with great versatility and robustness in noisy data filtering. EEMD consists of sifting and making
an ensemble of a white noise-added signal, and treats the mean value as the final true result. This is due to the use of a
finite, not infinitesimal, amplitude of white noise which forces the ensemble to exhaust all possible solutions in the sifting
process. These steps collate signals of different scale in a proper intrinsic mode function (IMF) dictated by the dyadic
filter bank. As EEMD is a time–space analysis method, the added white noise is averaged out with a sufficient number
of trials. Here, the only persistent part that survives the averaging process is the signal component (original data),
which is then treated as the true and more physically meaningful answer. The main purpose of adding white noise was
to provide a uniform reference frame in the time–frequency space. The added noise collates the portion of the signal of
comparable scale in a single IMF. Image data taken as time series is a non-stationary and nonlinear process to which the
new proposed EEMD method can be fitted out. This paper reviews the new approach of using EEMD and demonstrates
its use on the example of image data analysis, making use of some advantages of the statistical characteristics of white
noise. This approach helps to deal with omnipresent noise.

Keywords: EmpiricalModeDecomposition (EMD),EnsembleEmpiricalModeDecomposition (EEMD), image analysis,
sifting, Noise-Assigned Data Analysis (NADA).

1 Introduction
EmpiricalModeDecomposition(EMD)hasbeenpro-
posed as an adaptive time-frequency data analysis
method. It has been proved in applications for ex-
tracting signals from data generated in noisy non-
linear and non-stationary processes [1, 2]. However,
there are still several known unresolved difficulties
with EMD. The first major weakness of the origi-
nal EMD is the frequent occurrence of mode mix-
ing, which is defined as a single IntrinsicMode Func-
tion (IMF). IMFs can also consist ofwidely disparate
scales, or can consist of a similar signal residing in
different IMF components. Mode mixing is often a
consequence of an intermittent signal, as discussed
by Huang et al. [2, 3, 4]. An intermittent signal can-
not only cause serious aliasing in the time–frequency
distribution, but can also make the physical mean-
ing of individual IMFs seriously unclear. To alle-
viate this drawback, Huang proposed the intermit-
tence test [3, 2, 4]. This test aimed to avoid sev-
eral difficulties. However the test caused its own is-
sues, which were also illustrated by Huang and Wu.
The first issue is the subjectively selected scale of
the test. By this intervention EMD became totally
adaptive. The second issue concerns separable and
definable data selection from timescales, which was

discussed by Huang and Wu et al. [4]. To overcome
the scale separation issue without introducing a sub-
jective intermittence test, Huang andWuproposed a
newNoise-AssignedDataAnalysis (NADA) method,
knownasEnsembleEMD(EEMD),whichdefines the
true IMFcomponents as themeanvalue of an ensem-
ble of trials [2]. Each trial consists of the signal plus a
white noise of finite amplitude. Binding white noise
helps to better cover real cases, as is common in cur-
rent papers on image data analysis [1]. Addingwhite
noise has shownEMD to be an adaptive dyadic filter
bank. Tomake suchan improvement,WuandHuang
were inspired byFlandring andGledhill and their re-
search in adding white noise to data analysis, which
helps to improve EMD results [5, 4]. The main prin-
ciple of EEMD was defined simply: “the added white
noise would populate the whole time–frequency space
uniformly with the constituting components of differ-
ent scales. When a signal is added to this uniformly
distributed white background, the bits of signal of dif-
ferent scales are automatically projected onto proper
scales of reference established by the white noise in
the background” [4, p. 2].
This approach was adopted on the basis of the

following observations:
1. A collection of white noise cancels itself in the
ensemble mean if averaged in a time domain;

49



Acta Polytechnica Vol. 50 No. 6/2010

therefore, only the signal can survive andpersist
in thefinalnoise-addedsignal ensemblewhenav-
eraged.

2. Finite, not infinitesimal, amplitude white noise
is needed to force the ensemble to exhaust all
possible solutions. Finitemagnitudenoisemakes
the different scale signals reside in the cor-
responding IMF, dictated by the dyadic filter
banks, and renders the resulting ensemble mean
more meaningful.

3. The true and physically meaningful answer to
EMD is not an answerwithout noise. It is desig-
nated tobe the ensemblemeanof a largenumber
of trials consisting of the noise-added signal.
The proposed EEMD method utilizes many im-

portant statistical definitions of noise [4]. The follow-
ing sections describe a part of the research done by
Huang and Wu on the relation between white noise
and a real signal. Image data is used in this paper.
Section 2 provides a brief introduction to Ensemble
Empirical Mode Decomposition, and several details
of drawbacks associated with mode mixing are de-
scribed. Section 3 describes the usefulness and ca-
pability of EEMD through examples in Image Data
Analysis and Optical Flow Assignment.

2 Introduction to Ensemble
Empirical Mode
Decomposition over EMD

As an introduction to a more detailed description of
the EEMD method, we begin with a short review
of EMD [2, 6]. The EMD method is an adaptive
method, with the decomposition based on data and
derived from data. In the EMD approach, the data,
time series x(t), is decomposed in terms of IMFs, as
has been described in [4] by Huang,

x(t)=
n∑

j=1

cn + rn

where rn is the residue of the original data x(t) and
n is the number of steps for extracting IMFs. IMFs
are simple oscillatory functions with varying ampli-
tudes and frequency. The extracted IMFs have the
following properties:
1. Throughout the length of a single IMF, thenum-
ber of extremes and thenumber of zero crossings
must either be equal or must differ at most by
one (although these numbers can differ signifi-
cantly for the original data set).

2. At any data location, the mean value of the en-
velope defined by the local maxima and the en-
velope defined by the local minimum is zero.

Fig. 1: The sine wave signal with three crests is used as
an example in an introduction to EEMD (Fig. 2)

Fig. 2: The first step of the sifting process. Panel (a)
is the input. Panel (b) identifies local highs (red dots).
Panel (c)plots theupperenvelope (upper reddashed line)
and the lower envelope (lower red dashed line) and their
input signal (blue line), and panel (d) are the difference
between the input and the mean mi(t) of the envelopes

In common practice, EMD is implemented
through a sifting process using only local extrema [2,
6]. Foranydata rj−1 the followingprocedureapplies:
1. identify all the local extrema (a combination of
highs and lows) and connect all these local highs
(lows) with a cubic spline as the upper (lower)
envelope

2. obtain the first component h(t) by taking the
mean m(t) of the upper and lower envelopes

3. Treat h(t)= x(t)− m(t) as the data, and repeat
steps 1 and 2 as many times as is required until
the envelopes are symmetric with respect to the
zero mean under certain criteria.
The final h(t) is designated as cj. The complete

sifting process stops when the residue rn becomes a
monotone function fromwhich nomore IMFs can be

50



Acta Polytechnica Vol. 50 No. 6/2010

extracted. The process is stopped using a stoppage
criterion. Two types of stoppage criteria have com-
monly been used. The first type was used by Huang
in 1998. It is based on a Cauchy type of conver-
gence test. The test requires the normalized squared
difference between two successive sifting operations
defined as Eq. 1

SDk =

∑T
t=0 |hk−1(t) − hk(t)|

2∑T
t=0 h

2
k−1

(1)

to be small. If this squared difference SDk is smaller
than the desirednumber, the processwill be stopped.
Setting up the right SDk value seems tobeaverydif-
ficult task, because no acceptable definition is avail-
able [2]. The second criterion is to set up a pre-
selected S-number, which deals with other issues
about how to select the appropriate number. These
difficulties led authors [1] to develop a new approach
to obtaining IMFs from the acquired signal (data),
which will be described and applied below.
Based on the research of Fladrin and Gledhill,

Wu andHuang [4] used their results as a background
for improving the definition of the EMD method.
Bearing the definition of EMD in mind, they showed
that EMD behaves as a dyadic filter bank when
white noise is populateduniformly through thewhole
timescale or time-frequency space [4].
Their description postulated that the total num-

ber of IMFs in a data set is close to log2(N), with
N as the total number of data points. This fact en-
sures a total number of valid IMFs with difficulties
around. When the signal is not merged with pure
noise, some scales can be missing, and this involves
the total number of IMFs, which can be lower than
the expectation log2(N). Another issue is caused by
modemixing, as in the previousEMD.Anadvantage
of this approach is knowledge of the expected IMFs
expressed in N number.

Fig. 3: The intrinsic mode functions of the input dis-
played in Fig. 1(a)

2.1 A definition of mode mixing
based on EMD

Modemixing has been defined as any IMFconsisting
of oscillation of dramatically disparate scales. When
mode mixing occurs, IMF can cease to have a differ-
ent physical meaning by itself. The signal can sug-
gest an invalid physical representation. This known
drawback was mentioned by Huang in [2, 3, 4]. An
example of the sifting process is presented in Fig. 2,
and its decomposition is shown in Fig. 3, as illus-
trated by Huang and Wu [4]. The fundamental part
of the data is the sine wave with unit amplitude.
At the three middle crests of the sine wave, high-
frequency intermittent oscillations with amplitudes
close to 0.2 ride on the fundamental sine wave, in
panel (a) of Fig. 2. This signal is denoted as the test
signal. Panels (b) and (c) show the sifting process
of the identifying highs (lows), as described above.
Panel (d) displays the result, which is affected by
visible mode mixing. This is because one envelope
was a mixture of envelopes of the fundamental sine
waveandthe intermittent signal, leading toa severely
distorted envelope mean, as shown in Fig. 2 and
Fig. 4. Fig. 4 shows the main issue caused by ap-
propriate localization of the highs and lows. An un-
pleasant implication of mode mixing leads to disad-
vantages of the original EMD method with the stop-
page criterion, as described byHuang [1, 2, 4]. Mode
mixing is also the main reason why the EMD al-
gorithm is unstable (Fig. 3). Any small perturba-
tion may result in a new set of IMFs, as reported
by Gledhill [4]. Obviously, the intermittence pre-
vents EMD from extracting any signal with similar
scales. EEMD [4], which will be briefly described
in the following sections, has been proposed to solve
these problems.

Fig. 4: Test signal (red dashed line) and the locations of
highs (blue circle) and lows (green circle)

51



Acta Polytechnica Vol. 50 No. 6/2010

2.2 Ensemble empirical mode
decomposition

As shown in theprevious test signal, the data (Fig. 2,
Fig. 3) comprises collections of the main signal and
some noise. To improve the precision of the mea-
surements, the idea of the ensemble mean becomes
a powerful approach. Data is collected by separate
observations, each ofwhich contains a differentwhite
noise realization [4].
To generalize this ensemble idea, the noise is pop-

ulated to a single data set x(t). The added white
noise is treated as possible random noise that would
be encountered in the measurement process. Under
such conditions, the equation of any observation can
be written as

Xt(t)= x(t)+ wi(t) (2)

In thisway, adifferent realizationofwhite noise wi(t)
will be added when there is only one observation.
(Eq. 2)
We will now make a brief review of the proper-

ties of EMD before proceeding to a more detailed
description of the EEMD method:
1. EMD is an adaptive data analysismethod based
on local characteristics of the data, and it there-
fore captures nonlinear, nonstationary oscilla-
tions more effectively

2. EMD is a dyadic filter bank for any Gaussian
white noise-only series

3. when the data is intermittent, the dyadic prop-
erty is often compromised in the original EMD,
as shown in the example in Fig. 3

4. adding noise to the data canprovide auniformly
distributed reference scale, which enables EMD
to repair the compromised dyadic property

5. the corresponding IMFs of different series of
white noise have no correlationwith each other.
Therefore, the means mi(t) of the corresponding

IMFs of different white noise series are likely to can-
cel each other.
Bearing in mind all these properties of the EMD

method, the proposed EEMD has been developed in
the following way:
1. add white noise series to the targeted data;
2. decompose the datawith addedwhite noise into
the IMFs;

3. repeat step 1 and 2, based on the realization of
different white noise series each time;

4. perform steps 1 and 2 until the IMFs of the data
set are close to log2(N), with N as the number
of total data points;

5. obtain the (ensemble) means of the correspond-
ing IMFs of the decompositions as the final re-
sult.

Fig. 5: Test signal with white noise added (red dashed
line) and the location of its highs (blue circle) and lows
(green circle)

Fig. 6: The input top panel, its intrinsic mode functions
(C1–6), and the trend (R). In panel C5, the original in-
put is plotted as the bold dashed red line, for purposes of
comparison

The main effect of decomposition using EEMD
is that the added white noise series cancel each
other in the final mean of the corresponding IMFs.
This means that the IMFs stay within the natural
dyadic filter windows, and thus significantly reduce
the chance of mode mixing and preserve the dyadic
property.
An example of the use of the EEMD method is

illustrated in Fig. 6, in a similar meaning as for the
previous EMD in Fig. 2 and Fig. 3. Clearly, the fun-
damental signalC5 is representedalmostperfectly, as
are the intermittent signals, if C2 and C3 are added
together. The fact that the intermittent signal ac-
tually resides in two EEMD components is due to
the average spectra of the neighboring IMFs of white
noise overlapping, as revealed by Wu and Huang. It

52



Acta Polytechnica Vol. 50 No. 6/2010

is necessary to combine the two adjacent components
to one IMF. The need for this type of adjustment
is easily determined through an orthogonal check.
Whenever two IMF components become grossly non-
orthogonal, we should consider combining the two
components to form a single IMF component [4].

3 EEMD method
implemented in image data
analysis

The previous theoretical example shows the main
concept of NADA, using the EEMD method as a
tool. Based on previous text discussion, the question
arises how the method could be used in image data
analysis. Every frame of an image comprises a time-
sequence of data that is processed by the brain. Our
brain gives an appropriate meaning to this ensemble
of data. In this way, synapses can work as a dyadic
filter bank. Data can be and mostly is affected by
all kinds of noise. There can be various reasons for
this noise. Examples are short-sightedness or day-
light. The observed object is not sharply displayed
on our retina and the data is treated in a damaged
way. This can affect itsmeaning or its values. In this
section, we will outline EEMD usage and how it can
be used in analyzing such data. An example of cube
rotation is displayed in Fig 7.
The brightness of the pixels over two images of

the same cube in line height 120 pixels forms the
test signal. The general size of the two images is
320×240 pixels in gray scale colormode. Gray scale
mode was chosen for the sake of simplicity to better
understand themean value. In the first step, the two
images were analyzed in a similar way. The line val-
ues were treated and the IMFs were extracted. As
the deviation, the amount of white noise was set to
0.2 and the EEMD ensemble number was set to 50.

Fig. 7: Cube rotation: the cube rotates fromamisaligned
position to frontal position. Theblackarrow indicates the
direction of rotation

Fig. 8: Cube 1 – signal line

Fig. 9: Cube 2 – signal line

Fig. 10: Cube 1 – All IMF functions extracted by EEMD

53



Acta Polytechnica Vol. 50 No. 6/2010

In theCube 1 trial, the input (the original bright-
ness data) is a visible jump around a width value
of 100, which corresponds with the edge of the real
cube (Fig. 7). Cube 1 is slightly dipped to the left
side from the center, as shown in the original image
(Fig. 8). The next peak inFig. 10 ofwidth of around
118 units can represent the second cube edge, which
is coherent with reality. Consequent perturbation in
thefirst IMFcan reflect the footmark. Thefirstmain
point here is the edge projection as a short pertur-
bation with definable peaks over a zero value. Other
IMFs can represent surface deviances. The last line
is the residuum of the data. The residuum is mostly
explained as a trend. The trend explanation can also
be used for our case, because the last line shows a
background change caused by the box. The original
data defines the camber in the real scope. This jump
is performed in the same way as in the trend line.
Only eight different IMFs were extracted, which cor-
responds to the EEMD basics of the finite data set
explained above. The data forCube 2 (Fig. 9) is also
treated and displayed in a similar sense as Cube 1.
The image of Cube 2 is taken from the front, which
is also visible in the original. The data jump and the
additional pit are smaller. This fact is caused by the
projection of Cube 2, which is logical. The surface
texture is alsodisplayed in the originalfigure. Aswas
described above, the EEMD method works with the
IMF summary.

Fig. 11: Cube 2 – All IMF functions extracted byEEMD

Fig. 12: Comparison between the original data set and
the IMF summary

All IMFs should have no mutual correlation.
IMFs should be independent of white-noise trials.
This means that a summary of all IMFs should give
us the original signal (Fig. 12). Finally, when we
make a comparison of the two cubes, we find some
kind of data shifts in the inputs and IMFs.
These shifts are treated by cube rotation from

right to left, as indicated above (Fig. 7). There is
no doubt that the two original figures are affected
by different kinds of noise. The noise was treated
during capture time. EEMD enables us to extract
data from images properly, even in different layers.
The main intention in edge detection is to find ap-
propriate differences over all image layers, which is
enabled by the use of the EEMD method presented
here. Another important point is the logically dif-
ferent image width of cube rotation. Fig. 13 points
out several areas where the cube edge can be pro-
jected on to the specific IMFs. Possible edge fluctu-
ations are marked in subplot C1. These fluctuations
are displayed as second fluctuations, because of the
main transition. The first and the last fluctuations
directly indicate a background change to/from the
cube. Now we see that we have two limited areas
where everything can be projected. In C2, the IMFs
still show similar behavior over cubes with a logical
width shift. The underlying change is shown in IMF
C3. The Cube 1 line is projected with no underly-
ing fluctuations, and it corresponds to realitywithno
additional cube edge. In Cube 2, however, the line
fluctuation reaches a smaller peak and some kind of
promontory is treated. This can be explained as a
cube surface projection. When we look back to the
original images, we see the true reality (Fig. 8 and
Fig. 9). We consider here that C3 corresponds with
the surface of the object where no inner edge fluctu-
ation is displayed. This assertion is not proven, and

54



Acta Polytechnica Vol. 50 No. 6/2010

Fig. 13: Comparison of cube EEMDs — the colored cir-
cles indicate important edge detection areas

Fig. 14: Cube 1 — A comparison between EMD and
EEMD methods and instability in analysis

will be a topic for further study. This result can be
taken as a valid conclusion here. In our analysis of
theCube 1 signal, we found three edges. In the anal-
ysis of the Cube 2 signal we found only two edges,
which correspond to the front view. The retrieved
result is considered to be the cube rotation from the
left dipped state to the right frontal view.

4 A comparison between
EMD methods and EEMD
methods

Fig. 14 presents a comparison between the original
EMDmethod and the newEEMDmethod, using im-
age data analysis.

5 Conclusion
A major drawback of EMD is the instability of the
method, see Fig. 14. The EMD method has a prob-
lem evenwith edge detection. Edge fluctuations over
the signal are more widely spread and are not as
strict as inEEMD.EMD lines showa strongmixture
of modes, where lines dramatically change direction
with no valid reasons. Any edge projected on to the
retina is awidth limited area,which causes strict sen-
sor stimulations. This assertion is closer to EEMD
behavior, where narrower fluctuations are depicted.
In brief, results produced by the EMD method

can be less valid thanEEMD lines, which offermuch
stricter behaviors. This instability can have a dra-
matic effect on further studies of any signal, not nec-
essary from an image [1, 2, 4, 6]. The reason for the
drawbackis linkedwith the ideaof acleardata signal.
This ideadoesnot correspondto realworldexamples.
The EEMD method uses a white noise-added signal.
This approach exhausts all possible solutions during
the shifting process (Fig. 4, Fig. 5). The comparative
test of the instability of the twomethods displays the
mainbenefit ofusing theEEMDmethod. The ability
todivide the incoming signal data intomore indepen-
dent IMFs functions and then analyze it gives power
to the EEMD method, where this signal, not nec-
essary an image, can be interpreted and processed
by layers. The natural fact of the independence of
IMFs provides an opportunity to treat any signals as
a summary of their layers, and this could be helpful
in future medical research. There is potential and
motivation for improving the proposed method.

Acknowledgement

This work was supported by the Grant Agency of
the Czech Technical University in Prague, grant
No. 2B06023 and by CTU grant SGS OHK2-021/10.

References
[1] Kopecký,M.: Hilbert-HuangTransformationand
its application in optical flow evaluation, STC,
2009.

[2] Huang,N.E., Samuel, S. P. Shen: Hilbert-Huang
Transformand ItsApplication,Politecnico diMi-
lano, Vol. 25. ISSN 0262-8856.

55



Acta Polytechnica Vol. 50 No. 6/2010

[3] Huang, N. E., Shen, Z., Long, S. R., Wu, M. C.,
Shih, E. H., Zheng, Q., Tung, C. C., Liu, H. H.:
The empirical mode decomposition method and
the Hilbert spectrum for non-stationary time se-
ries analysis,Proc. Roy. Soc. London 454A, 1998,
p. 903–995.

[4] Wuand, Z., Huang, N. E.: Ensemble Empiri-
calModeDecomposition andNoise-assistedData
Analysis Method, World Scientific Publishing
Copany, Advances in Adaptive Data Analysis,
Vol. 1, No. 1 (2009), p. 1–41.

[5] Flandrin, P., Goncalves, P., Rilling, G.: EMD
equivalent filter banks, from interpretation to ap-
plications, in Hilbert-Huang Transform: Intro-

duction and Applications, eds. N. E. Huang and
S. S. P. Shen, World Scientific, Singapore, 2005.

[6] Kokeš, J.: Okamžitá frekvence a Hilbert-
Huangova transformace. Sborńık konference In-
teligentńı systémy pro praxi, Lázně Bohdaneč,
2006. ISBN 80-239-6535-2.

Miroslav Kopecký
E-mail: Miroslav.Kopecky@fs.cvut.cz
Department of Instrumentation
and Control Engineering
Czech Technical University in Prague
Faculty of Mechanical Engineering
Technická 2, 166 27 Praha, Czech Republic

56