Empirical Mode Decomposition in data-driven fMRI analysis
John McGonigle
Department of Computer Science
University of Bristol
Bristol, UK
[email protected]
Majid Mirmehdi
Department of Computer Science
University of Bristol
Bristol, UK
[email protected]
Abstract—Empirical Mode Decomposition has emerged in
recent years as a promising data analysis method to adaptively
decompose non-linear and non-stationary signals. Here we
introduce multi-EMD, to be used where there are many
thousands of signals to analyse and compare, such as is common
in the analysis of functional neuroimages. The number of component signals found through Empirical Mode Decomposition
varies at each location in the brain. We seek to rearrange
these components so that they may be compared to others
at a similar temporal scale. This is a data-driven process
based on grouping those components which have similar
dominant frequencies to target frequencies which have been
found to be most common from the initial decomposition. This
new set of rearranged components is then clustered so that
regions behaving synchronously at each temporal scale may be
discovered. Results are presented for both simulated and real
data from a functional MRI experiment.
Keywords-Magnetic resonance; Signal resolution; Timefrequency analysis; Clustering methods;
I. I NTRODUCTION
Since its development by Huang in 1998 [1], Empirical
Mode Decomposition (EMD) has been applied across many
diverse disciplines as a useful technique in signal decomposition. These have included tomographic reconstruction
in PET [2], diagnosis of mechanical problems in rotating
machinery [3], examination of EEG data [4] and bandpass
filtering [5]. Specifically, its adaptive nature is useful in a
data-driven analysis since no filter size or basis function is
specified beforehand, but rather discovered from the data
itself. We propose to use this technique as one step in a
data-driven analysis of functional neuroimages.
Functional magnetic resonance imaging (fMRI) is a form
of functional neuroimaging which takes many 3D scans
of the human brain over time. Each spatial position will
therefore have several hundred samples, which will vary in
intensity due to factors such as blood oxygenation, slight
subject motion, or artefacts inherent in the imaging process.
The resulting data will therefore be made up of many
thousands of these time courses. The strength and temporal
characteristics of these signals is of interest since they enable
us to elucidate and localise neural behaviour in the brain.
A wide variety of analysis techniques are applied to this
type of data. The most popular of these uses the general
Andrea L. Malizia
Psychopharmacology Unit
University of Bristol
Bristol, UK
[email protected]
linear model to identify voxels whose time course correlates
with the a priori experimental procedure, implemented as
statistical parametric mapping (SPM) [6]. Other techniques
attempt to group voxels which exhibit similar behaviour.
In wavelet-based cluster analysis (WCA), coefficients from
wavelet decompositions are grouped through optimisationbased clustering [7]. One drawback of decomposition methods such as that used in WCA is that the number and
temporal resolution of each component is defined a priori.
As such, there may be non-ideal situations where a true
component is smeared across several different frequency
ranges, or temporal scales.
In EMD, a signal may be adaptively decomposed into a
number of component signals, where, in the case of fMRI,
this number will vary across the many thousands of voxels
in the brain. This creates a difficulty when attempting to
compare decomposed signals from different voxels. Here
we propose a solution to this problem by the introduction
of multi-EMD and explore its use as a non-supervised and
completely data-driven decomposition step, splitting the data
into a number of different components, each corresponding
to a temporal scale discovered from the data itself before
carrying out clustering at each of these scales. It is applied
here on simulated and real data from an fMRI experiment.
II. P ROPOSED M ETHOD
A. Empirical Mode Decomposition
Empirical Mode Decomposition, as introduced in [1],
attempts to decompose a signal into a finite and often low
number of component signals or Intrinsic Mode Functions
(IMFs). These IMFs are defined as having (i) the same number of zero crossings as they have extrema (or a difference
of 1) and (ii) symmetric envelopes defined by local extrema.
Following [1] we describe the sifting process used to
discover each IMF:
1) Taking the original signal X, find all local extrema.
Connect all local maxima by a cubic spline to form an
upper envelope. Repeat for all local minima to form a
lower envelope.
2) Take the mean of these envelopes, m1 . The difference
between X and m1 is h1 , which we regard as a protoIMF.
non-stationary signals. Figure 1 shows an example of the
decomposition of a signal into its IMFs and residual.
Original Time Course
4
0
B. Use in brain imaging
−4
IMF 1
2
0
−2
IMF 2
Amplitude
2
0
−2
IMF 3
2
0
−2
IMF 4
2
0
−2
IMF 5
2
0
−2
Residual
2
0
−2
0
50
100
150
Timepoints
200
250
Figure 1.
An original voxel time course (from fMRI of the human
brain) together with its IMFs and residual following Empirical Mode
Decomposition.
EMD has been predominantly used on one signal at a time
and often with supervision. In fMRI one often has many tens
to hundreds of thousands of signals, or voxel time courses, to
examine. As in WCA, it can be useful to examine this data
at a suitable temporal scale, that is, one is often interested
in looking at the data without its higher frequency noise or
lower frequency drift. However, the number of component
temporal scales each signal is broken down into in WCA
will depend on the basis function imposed by the user.
By using EMD to perform a decomposition, this number
of component temporal scales and their frequency content
is adaptively driven by the data itself. Indeed, one IMF
may contain several different frequencies. This creates the
situation whereby different voxel time courses are broken
down into different numbers of IMFs. Unlike WCA, it is no
longer possible to choose just one or more temporal scales to
cluster the data at, since, for example, the 3rd IMF from one
voxel may be more comparable to the 4th IMF from another.
This will often depend on the signal conditions which will
vary spatially in the brain.
C. multi-EMD
3) If h1 does not fulfill the properties of an IMF steps
1 and 2 are carried out on it, so that h11 = h1 −
m11 . This continues for k iterations until h1k satisfies
the properties of an IMF or the stopping criteria are
reached (see below).
4) After the first IMF is found, it is subtracted from
the original data, and this remaining signal, X − h1k ,
treated as the original data in a repeat of steps 1 to 4.
5) This is all carried out n times, finding a component
signal, IMFj , at each level until no further IMFs can
be discovered and only the residual, r, remains.
The original data may therefore always be reconstructed
from the IMFs and residual as
X=
n
X
IMFj
+ r.
(1)
j=1
So that the process will not carry on indefinitely a
stopping criteria may be used for those proto-IMFs that do
not satisfy the definition of an IMF. Also following [1],
this can be achieved by limiting the size of the standard
deviation, SD, found from two consecutive sifting results as
PT
|hk−1 (t) − hk (t)|2
.
(2)
SDk = t=0PT
2
t=0 hk−1 (t)
Since the decomposition is based on the local characteristics of the data it may be applied to non-linear and
If the original data is to be broken down into a number
of temporal scales, it is necessary to discover what this
number should be, and also which IMF or residual from
each voxel should be used to represent the data at each
scale. To achieve this we introduce the Empirical Mode
Decomposition of multiple signals, or multi-EMD. This is
carried out as follows:
1) Perform EMD at every voxel, producing n IMFs for
each.
2) Find the median number of IMFs plus one (for residuals) that the data has been decomposed into. This is the
number of multi-EMD components, n0, to rearrange
the IMFs into.
3) Find the median most powerful frequency across each
of the initially decomposed IMF levels, and use these
as the target frequency for each multi-EMD component.
4) A new data-matrix is constructed for each multi-EMD
component and filled, at each voxel, with the IMF
or residual from that spatial position which has a
dominant frequency closest to the target for that scale.
K-means clustering may then be performed on any or all
of these new multi-EMD components.
D. Experimental set-up
1) Phantom: The phantom data used here is based on
that in [7]. It is a single square slice of side 128 with each
pixel having 256 timepoints (Figure 2(a)). Each row contains
B. Phantom
(a) One frame of the
phantom data (at a
CNR 5 times that used
here for ease of visualisation).
Figure 2.
(b) The number of
IMFs
each
pixel
may be decomposed
into. Darker colours
represent fewer IMFs.
(c) K-means using 5
clusters on the undecomposed phantom
data.
Different views of the simulated phantom data
circles, of radius 9 pixels, of a sine wave at decreasing
1
1
1
1
, 8π
, 12π
and 16π
Hz),
frequencies (from top to bottom 4π
with the columns being this signal with a contrast-to-noise
ratio (CNR) of 1, 0.6, 0.5 and 0.25 from left to right. These
are surrounded by a square area of side 14 containing a
Gaussian taper of the circles’ signal. This is the equivalent to
signals with maximum percentage changes of 2%, 1.2%, 1%
and 0.5% respectively. The background contains Gaussian
noise with a standard deviation of 2% signal change. These
are the type of noise levels one can expect to see in fMRI.
2) Functional MRI: The experimental data used here
comes from a block design sensorimotor task which involved
bilateral finger tapping, audio tones and a flashing checkerboard, all in synchrony at 3Hz. The data was collected as part
of a site inter-variability experiment, of which one subject’s
data from one day is examined here [8]. Total time was
85s with a repetition time of 3s. Each block was 15s long,
interleaved with a control condition of staring at a cross for
the same period. This design would be expected to show
activity in the motor, visual and auditory areas.
3) Wavelet-based Cluster Analysis: The most similar
related method to that which we are presenting here is the
wavelet-based cluster analysis (WCA) of Whitcher et al.
[7]. It seeks to decompose voxel time courses using Haar
wavelets so that whole frequency scales may be removed
from the data and only those frequency scales of relevance to
the experimental design retained for an optimization-based
clustering step. Here we will perform WCA on our phantom
data as a comparison to our multi-EMD method.
1) Pure k-means: The results of k-means clustering on
the undecomposed phantom data can be seen in Figure
2(c) (in all the clustering results shown here k-means was
used with 5 clusters). Only those pixels which have their
signal at a CNR of 1 and 0.6 are sensibly grouped, while
those pixels with a CNR of 0.5 and 0.25 are clustered with
the background noise, the second signal type cannot be
distinguished at all.
2) WCA: In Figure 3 the results of clustering at each level
of wavelet coefficients can be seen. At each step the effective
frequency is halved along with the temporal resolution.
Although the lower CNR signals are better discerned than
without any decomposition, it is clear that most scales retain
signal which would not be expected. In effect, information
from the single frequency simulated signals is smeared
throughout all the levels of wavelet coefficients.
3) multi-EMD: In Figure 4 the results of our multi-EMD
based clustering can be seen. Our algorithm has broken the
data up into 7 different components which have the target
frequencies of 0.4, 0.18, 0.098, 0.058, 0.035, 0.020 and
0.011 Hz. The simulated signals have the actual frequencies
of 0.080, 0.040, 0.026 and 0.020 Hz. It is seen that there is
a better discrimination between temporal scales when using
multi-EMD than with WCA. In a perfect decomposition of
the data only one row would be visible in each component
image. In effect, this EMD based decomposition is similar
to an adaptive and data-driven bandpass filter.
C. Functional MRI
The fMRI data was broken down into 5 different components, with target frequencies of 0.12, 0.055, 0.031, 0.020
and 0.0078 Hz. The frequency of the onset of each block
of activity in the experimental design was 0.033 Hz, corresponding to the 3rd component found by our algorithm
(shown in Figure 5). The highest percentage signal changes
are contained withing the green and blue clusters, while
other voxels behaving in synchrony but of lesser magnitude
(though still having a range above 0.5%) are shown in red.
Slices 12, 20 and 28 show activity corresponding to the
visual, auditory and motor cortex respectively as could be
expected from the experimental design.
IV. D ISCUSSION
III. R ESULTS
A. Number of IMFs
The total number of IMFs that make up a signal can be
a useful relative measure in its own right of whether it is
of possible interest or merely represents background noise.
In Figure 2(b) it can be seen that those areas of the image
where signals have been encoded can be decomposed into
fewer IMFs. This is to be expected since the less complex
a signal is the fewer component signals will make it up.
Due to the mixed modes in most IMFs, it may be useful
to perform a wider search for the best temporal scale,
incorporating the top 2 or 3 most powerful frequencies. This
would allow the process to be more resilient to aliasing from
other temporal scales. The technique could also be used
as part of an adaptive noise and artefact reduction step,
where groups of voxels found by the user to be due to
motion or other unwanted effects could be removed from
any further analysis, or have their signals used as unwanted
effects regressors in a later model driven analysis approach.
Figure 3. WCA applied to the phantom data. Each image is a different temporal scale. In a perfect decomposition each image would only show one row.
Effective frequency decreases from left to right in approximate powers of two. Colours do not correspond between scales.
Figure 4. The results of our multi-EMD based cluster analysis on the phantom data. The number of scales was chosen by our algorithm. An ideal temporal
decomposition would show only one row in each image. Colours do not correspond between scales.
A similar artefact removal process is often used in an
ICA based analysis. We have presented a novel approach
to the analysis of large groups of signals such as those
found in fMRI. Our method performs the Empirical Mode
Decomposition on many signals followed by the selection
and grouping of individual Intrinsic Mode Functions by
how similar their most powerful frequencies are to target
frequencies (themselves found from the data). This ensures
that the most ideal IMF from each signal is chosen since
different signals will often have a varying number of IMFs.
In brain imaging this will often be the case where separate
regions may have the same underlying neural response, but
experience different noise conditions. The process is datadriven, splitting the original data up into several components,
each at a different temporal scale. Further optimisation
based clustering may then be carried out on the rearranged
data. As such, multi-EMD based clustering appears to be
a promising approach in an exploratory and data-driven
analysis of functional brain imagery.
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Figure 5. Several slices from the 3rd multi-EMD scale. The algorithmically
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