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Selection of Wavelet Based Optimal Denoising
Method in fMRI Signals
Cemre Candemir
International Computer Institute, Ege University, 35100, Izmir, Turkey
cemre.candemir@ege.edu.tr
Abstract— Functional magnetic resonance images (fMRI)
contains high amount of noise due to their structures. This high
noise limits the correct interpretation of the information
contained in the signal. In order to interpret the data correctly,
it is necessary to eliminate the noise with effective and efficient
noise reduction techniques while protecting the information in
the signal. The aim of this study is to find an optimal
methodology for denoising. For this purpose, Cubic Spline,
Discrete Wavelet Transform (DWT) and Maximal Overlap
Discrete Wavelet Transform (MODWT) are used as wavelet
based noise reduction methods. These methods are applied on a
real fMRI signal and compared with different parameters and
their performance was evaluated.
Keywords — fMRI; denoising; wavelet transform; DWT;
MODWT; cubic splines.
I. I
NTRODUCTION
Functional Magnetic Resonance Imaging (fMRI) is a non-
invasive neuroimaging tool that widely used to detect and
evaluate the brain activity. In fMRI, a volumetric image
consisting of a series of images is usually obtained, with time
intervals of 2 or 3 seconds. Mainly, fMRI focuses on the
change in the amount of oxygen levels in the blood transported
to the region where the brain activates in relation with the
stimulus rather than the tissues. This signal which the MRI
device focuses on is called the Blood Oxygen Level
Dependent (BOLD) signal. During fMRI scan, the brain is
displayed in slices and each slice contains an average BOLD
signal value in a given volume called voxel. Signal values are
obtained separately for all voxels in each slice. These values
also produce a time series for each voxel [1]. The active
regions are determined by statistical analysis of these time
series. Detection and characterization of the changes
depending on stimuli is the main objective of fMRI time
series analysis.
However, even though theoretically all BOLD signals can
be expressed with the same mathematical model, the obtained
actual signals can be varying in complex waveform. Unlike
theoretical signals, real signals are also affected by various
other physiological features, such as head movement and/or
breathing. These movements do not reflect any neural
activity, rather they cause fluctuations in the BOLD signal
and generate noise on the signal, resulting in low signal-to-
noise ratio (SNR). Low SNR restricts the interpretation of the
information within the BOLD signal, and in some cases even
causes the noise to be regarded as the actual signal. The main
purpose of suppressing noise on BOLD signals is to minimize
the noise from the observed signal without generating a
change on the actual signal characteristic that involves neural
activation.
Compared to methods such as EEG or MEG, fMRI
appears to have a weaker SNR despite a better spatial
resolution. This requires the use of efficient and effective
noise reduction techniques to smoothen the functional data.
Typically, Gaussian Smoothing is used for noise removal on
BOLD signals. However, if the radius of the applied Gauss
filter is much larger than the area that shows activation (or is
expected to activate), it can also reduce the significance of the
signals by including the surrounding voxels in the filter [2].
Considering the related literature, it is seen that methods such
as Independent Component Analysis (ICA) [3], Wiener
Filtering [4], Harmonic model [5] are recommended for noise
removal. Although most of these methods are working on 2-
dimensional images and they aim to increase the
determination performance of the activation areas.
In this study, an efficient and optimal denoising
methodology has been studied to find, which reduces the
noise intensity on the fMRI time series while at the same time
does not cause any data loss on the signal. Cubic Spline,
Discrete Wavelet Transform (DWT) and Maximal Overlap
Discrete Wavelet Transform (MODWT) are used as noise
reduction methods. Performance comparisons are made with
different parameters of the three methods, depending on the
timing of the stimuli given in the fMRI experiment, reflecting
the activation times in the BOLD signal. Whit this study, it is
aimed to present a reliable method for low SNR BOLD
signals obtained from complex fMRI experiments in which
activation times vary widely.
II.
DATA SET AND ANALYSIS
A. fMRI Experiment and Acquisition
In fMRI experiments, the participant lying in the MRI
scanner is asked to perform a specific task simultaneously
while performing a series of brain scans. With these tasks, by
interpreting the changes in the BOLD signal with statistical
analysis, the brain regions showing local signal change, in
other words activated regions, are detected. The fMRI dataset
used in the study was obtained by SoCAT research group
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(Standardization of Computational Anatomy Techniques for
Cognitive and Behavioral Sciences: http://socatlab.com/) by
healthy and voluntary participants using Siemens 3T
Magnetom scanner. Participants completed the motor task
during the fMRI scan. The fMRI task that forms the dataset
is a finger movement experiment in block design structure
consisting of rest and movement sections that follow each
other in order to determine the areas related to the motor
functions of the brain. During 10 consecutive rests and 10
motor movement periods, the participants lie dormant for 60
seconds without performing any activity in the rest block,
while they continuously open and close their fingers for 60
seconds in the motion block.
B. Dataset and Preprocessing
The total length of the fMRI experiment is 1200 seconds.
The cycle time between the corresponding consecutive points
on the repeating series of pulses, called Repetition Time (TR)
is 2s, and time from the center of the first pulse to the center
of echo, called Echo Time (TE) is 30 ms. Flip Angle (FA) is
90o. The images are 53 x 63 in size and consist of 52 slices
and the size of each voxel is 3 x 3 x 3 mm.
First, the obtained functional data was converted to NIFTI
format. After that, by using the SPM.v12 tool which works
on MatLab®2017b, realign, slice timing, co-registration,
normalization, segmentation and smoothing preprocessing
steps were performed. Participants, who showed more than
normal movements, were excluded from the analysis. After
the 6-step preprocessing, all obtained images are mapped to
the MNI brain space and the voxel-to-voxel is registered.
These preprocessing steps are important notably on account
of representing the same coordinate for each voxel especially
in group analysis.
C. Data Model
The BOLD signal is modeled as the convolution of the
hemodynamic response function (HRF) with a given stimulus.
The BOLD signal of a voxel at the time is indicated by (1).
ݕ
ሺ߬ሻൌߚ
כݔሺ߬ሻ߳
ሺ߬ሻሺͳሻ
Here, ݕ
represents an ܰݔͳ dimensional vector which N
indicates the number of scans, ߚ
indicates the effect of the
coefficients of each x parameter and ߳
ሺ߬ሻ is the noise-induced
errors. These errors consist of with zero mean, independent
and identical distributed (i.i.d) with ߪ
ଶ
variance of random
normal variables ቀ߳ଓଓ݀
෪ܰሺͲǡߪ
ଶ
ሻቁ [6]. ݔሺ߬ሻ is modeled by the
convolution of s () and h () which indicates stimulus and
HRF, respectively (2).
ݔሺ߬ሻൌ ݏሺ߬ሻכ݄ሺ߬ሻ
(2)
GLM model of a voxel i of an fMRI experiment with ݆
parameters is expressed by (3)
ݕ
ൌߚ
ଵ
ݔ
ଵ
ߚ
ଶ
ݔ
ଶ
ߚ
ଷ
ݔ
ଷ
ڮߚ
ݔ
ߝ
ሺ͵ሻ
This model can be rewritten as ܻൌܺߚ߳ in matrix
format. Here, Y represents the BOLD signals of ܰݔͳ
dimensional vector of voxels; X denotes the design matrix
generated by each variable in the fMRI experiment. Since the
fMRI task aims the responsible areas form the activation, the
volume-of-interest (VOI) is determined as the motor regions,
which is Broadmann Area 4 (BA4) and Broadmann Area 6
(BA6) can be seen in Fig.1(a). The observed neural activities
are also coherent with the determined VOIs (Fig.1(b)) The
time series were modelled with the equation (3) and
participants are stimulated at the moments
w
(
l)
={30,90, 150,
210, 270, 330, 390, 450, 510, 570} for 30 TR periods.
III. D
ENOISING
M
ETHODOLOGIES
fMRI data are quite noisy structures in their nature. As we
aforementioned, the noise level may also increase during the
scanning by several reasons. During the analysis, this effect
is smoothed as possible using Gaussian kernel in
preprocessing steps, but the obtained signal still contains a
high level of noise.
The applied smoothing steps can suppress the noise as well
as the detailed features in the original images. Considering
this, a reliable denoising methodology should be applied in a
way that both suppresses the density variation of the noise
and not changing the features in the functional image.
A. Cubic Splines
Modeling the data with polynomials is a common method.
However, the usage of high-order and low order polynomials
may have some disadvantages. For example, the curves
produced with high order polynomials cause the modeling
error when the function is too closely fits to the data, i.e.
overfitting, while the curves generated with low order
polynomials may not be fitted enough to represent the data
set, i.e. underfitting. For this reason, the disadvantages of
high and low order polynomials were tried to be eliminated
with cubic splines. Instead of fitting a curve to the whole data
set, different curves were fitted to each interval between the
points (nodes) in the data set with the spline functions. Thus,
the spline interpolation can nicely match the smoothness of
the underlying function.
It is possible to obtain a continuous polynomial that is
suitable for the data, as the spline functions defined at every
point on the data and each range can be expressed with a
function that changes properly according to the structure of
the appropriate polynomials. A k
th
-degree spline is a
piecewise polynomial function which continuous at all
derivatives 1
st
,…,(k-1)
th
and the derivatives are also should be
defined at every interior point (ݔሻ.
(a)
(b)
Fig. 1. (a) Determined Voxel-of-Interest (VOI) at Left (blue) and Right
(red) hemispheres (b) activation areas that coherently found with the VOI
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Smoothing spline is a function which softens the f(x
i
)
function that obtained from a series of noisy y
i
observation
values, by minimizing the total squared curvature of the
spline in goodness of fit of the predicted f’(x
i
) function [7]. A
smoothing spline function is defined by (4).
൛ܻ
െ݂
መሺݔ
ሻൟ
ଶ
ߣන݂
መ
ᇱᇱሺ௫ሻ
మ
݀ݔ
ୀଵ
ሺͶሻ
Here 0 is the softening parameter. As approaches
zero, it moves away from softening to the interpolation
spline, and vice versa, it approximates the least squares
estimate.
The most commonly used smoothing spline is the third-
order cubic spline (k = 3) [8]. An f function, ݂ǣԹ՜ Թ which
passes through the set of points ݐ
ଵ
൏ڮ൏ݐ
ǡݐൌሺݔǡݕሻ, is
defined as a cubic spline if and only if it satisfies the
following conditions: [8]:
i. There is a cubic polynomial between each pair of
points ሺݐ
ǡݐ
ାଵ
ሻ.
ii. It is defined and continuous for all points.
iii. Its first and second derivatives are continuous at
every point.
A cubic spline is a 3
rd
-degree polynomial that provides all
of these conditions together and is defined by (5).
ܵ
ሺݔሻൌܽ
ሺݔെݔ
ሻ
ଷ
ܾ
ሺݔെ ݔ
ሻ
ଶ
ܿ
ሺݔെݔ
ሻ݀
ሺͷሻ
B. DWT
With wavelet transforms, the locations of the different
frequencies contained in the original signal and the points
associated with this frequency can be detected in a temporal
dimension. They are widely used in many different areas due
to their high frequency resolution at low frequency and high
time resolution at high frequencies.
Discrete Wavelet Transform (DWT) can successfully
analyze more complex problems as the analyzed signal
provides both frequency and position information. They work
more efficiently in the analysis of specific frequencies because
they reduce calculations compared to Fourier transforms [9].
Basically, the DW conversion of an x signal is calculated by
the convolution of a series of filters through which the signal
is passed at regular intervals (Eq, (6)).
ݕሾ݊ሿൌሺݔכ݃ሻሾ݊ሿൌσݔሾ݇ሿ݃ሾ݊െ ݇ሿ
ஶ
ିஶ
(6)
Here, g[n] and h[n] indicate the high and low pass filters
and n indicates the number of observations in the signal.
The discrete wavelet transform of a noisy BOLD signal
given by equation (3) is defined as the ܿ
ൌܹ
ௗ௪௧
்
Ǥݕ
where
ܹ is the orthonormal regression matrix.
C. MODWT
Maximal Overlap Discrete Wavelet Transform
(MODWT) is used to examine scale-dependent behaviors on
the signal. It is similar to DWT in that it is both a linear
filtering process creates a wavelet and scaling coefficient
based on time. However, unlike DWT, however, MODWT
is not orthogonal and can be defined for all sample sizes [9].
Basically, it is based on the use of the values that separated
by DWT decomposition at each level.
MODWT has several advantages over DWT. Since it
facilitates the alignment of the decomposed wavelet with the
original time series, MODWT provides an easy comparison
between the original and decomposed signal. In addition,
while DWT is dependent to the initial point of the series,
unlike DWT, MODWT is not affected by the beginning of
the time series [9]. As another advantage, the excessive
number of wavelet coefficients it contains increases the
degree of freedom and reduces the variance of the estimates.
And, MODWT can be used for random length time series.
As similar to DWT, the MODW transform of the BOLD
signal defined by equation (3) is defined as ܿ
ൌ
ܹ
ௗ௪௧
்
Ǥݕ
, where ܹ
ௗ௪௧
is the conversion matrix.
IV. A
PPLICATION AND
R
ESULTS
It is desirable that the signal-to-noise ratio of a BOLD
signal is as high as possible. However, the actual signal
obtained due to motion, physiological or device-related
reasons cannot be noiseless. In Figure 2, the visualization of
the noiseless theoretical signal (orange) which obtained
through the convolution of the stimuli (onsets and durations
depicted in black) with HRF and the actual BOLD signal
(blue) received from the motor-related regions of the brain
(BA4 & BA6) are given. It is obvious that the effect of the
noise actual BOLD signal contains can be clearly
distinguished even visually when compared with the
theoretical signal. When analyzed in terms of SNR values and
the strength of the noise affecting the signal, it is seen that the
original signal has a noise power of -22.3582 dbm while the
SNR value is 9.0038dB. These values are obtained in the
theoretical signal as 26.2468dB and -0.6889dbm
respectively. The difference between theoretical and actual
BOLD signals SNR values shows how effective the noise is
on the signals.
Cubic spline, DWT and MODWT methods were applied
with different parameters on the BOLD signals. The signal
characteristic after noise reduction and any change or loss in
the information it contains were tested by determining known
activation points in the denoised signal.
Smoothing parameters for the cubic spline on the BOLD
signal were selected as P = {0.2,0.4,0.6,0.8}. Change point
detection methods were also run on each of these four
smoothing levels. The average BOLD signal of a randomly
selected participant's VOI voxels and the noise-reduced
signal with cubic spline interpolation are given in Figure 3.
Fig.2. Visualization of the original BOLD signal (depicted in blue), stimuli
(depicted in black) and theoretical signal obtained (depicted in orange) by
the convolution of stimuli and HRF.
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While the signal converted with a cubic spline follows a
zero-mean time series, it is seen that the height of the signal
changes according to the magnitude of the activation in the
regions where there are changes, and the frequency of the
signal increases as the
softening parameter increases, which
makes it difficult to interpret the signal.
DWT and MODWT were performed with soft
thresholding using wavelet levels ݈ ൌ ሼʹǡ͵ǡͶǡͷሽ using Haar,
Daubechies and Symmlet wavelets. The signal obtained from
the average of VOI voxels and noise reduced signal by
applying DWT and MODWT are shown in Figure 4.
While at low level of decomposition (l) DWT and
MODWT converge similarly to the original signal, it can be
seen that MODWT gives more efficient results as the wavelet
level increases.
In Table I, as the result of the wavelet functions, the SNR
value of the signal and the value changes of the noise power
contained in the signal are presented. When compared the
performances of DWT and MODWT, it can be observed that
MODWT can produce higher SNR value than DWT and the
noise power is highly suppressed. The db6 function gives the
highest SNR value among the applied wavelet functions.
TABLE
I.
SNR
AND
NOISE
POWER
VALUES
IN
THE
BOLD
SIGNAL
AS
A
RESULT
OF
APPLICATION
OF
DIFFERENT
WAVEFUNCTIONS
haar db6 sym5
DWT
SNR (dB) 9.5158 15.6110 15.4366
Noise Powe
r
-22.946 -28.968 -28.800
MODWT
SNR (dB) 16.2043 31.4897 27.1087
Noise Powe
r
-29.640 -44.844 -40.462
V. C
ONCLUSION
The purpose of reducing the noise rate on the time series
of fMRI BOLD signal is to remove the noise as possible as
from the signal while keeping the desired piece of information
unchanged as well. For this purpose, wavelet transforms
which are easy, adaptive and effective can be used. These
methodologies were performed on real fMRI signals extracted
from the responsible area of motor movement in the human
brain. VOI was specified as BA4 and BA6. Performance of
different wavelet functions with different parameters were
also compared and evaluated according to the information of
activation onsets. The distinctive feature of all noise reduction
methods tested is the amount of signal smoothing. This feature
plays an important role in the applicability of the methods. The
results obtained from the study show that wavelet response is
directly related to wavelet function. It was shown that spline
functions can successfully settle on the signal, both frequency
and temporally, while daubechies and symmlets provide good
convergence to the signal.
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(a)
(b)
Fig. 3. The change of signal patterns at activation points after cubic spline
interpolation with (a) smoothing level P = 0.4 (b) smoothing level P=0.8.
(a) Haar
(b) Daubechies
(c) Symmlet
Fig. 4. Change of activation duration and activation points after DWT
and MODWT are applied with (a) Haar (b) Daubechies (c) Symmlet
function
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