Wavelet transform-based Wiener filtering of event-related fMRI data.

Abstract

The advent of event-related functional magnetic resonance imaging (fMRI) has resulted in many exciting studies that have exploited its unique capability. However, the utility of event-related fMRI is still limited by several technical difficulties. One significant limitation in event-related fMRI is the low signal-to-noise ratio (SNR). In this work, a method based on Wiener filtering in the wavelet domain is developed and demonstrated for denoising event-related fMRI data. Application of the technique to simulated and experimental data demonstrates that the technique is effective in reducing noise while preserving neuronal activity-induced response.

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Wavelet Transform-Based Wiener Filtering
of Event-Related fMRI Data
Stephen M. LaConte, Shing-Chung Ngan, and Xiaoping Hu*
The advent of event-related functional magnetic resonance im-
aging (fMRI) has resulted in many exciting studies that have
exploited its unique capability. However, the utility of event-
related fMRI is still limited by several technical difficulties. One
significant limitation in event-related fMRI is the low signal-to-
noise ratio (SNR). In this work, a method based on Wiener
filtering in the wavelet domain is developed and demonstrated
for denoising event-related fMRI data. Application of the tech-
nique to simulated and experimental data demonstrates that
the technique is effective in reducing noise while preserving
neuronal activity-induced response.
746–757, 2000. © 2000 Wiley-Liss, Inc.
Key words: event-related fMRI; denoising; stationary wavelet
transform; Wiener filter
Magn Reson Med 44:
In the past several years functional MRI (fMRI) (1–3) based
on the blood oxygen level-dependent (BOLD) (4–7) con-
trast has become a ubiquitously used technique for study-
ing brain function. Analogous to PET studies, most fMRI
studies have utilized “block design” experiments in which
a stimulus is presented for extended periods of time. In
this design, pixel intensities in images acquired during a
period of stimulus (and presumably response) are com-
pared to those from periods with no stimulus. This ap-
proach essentially provides a steady-state view of the neu-
ronal response at an acceptable signal-to-noise ratio (SNR).
Recent advances in both image acquisition and data anal-
ysis have made it possible to exploit the high temporal
resolution available in MRI to perform what is termed
“event-related” or “single trial” experiments (8). This new
experimental design, termed ER-fMRI hereafter, closely
parallels evoked response studies performed with high
temporal resolution techniques such as evoked response
potential (ERP) measurements. In these experiments, the
subject is presented with a brief stimulus or performs a
single instance of a task, and the corresponding transient
response is measured. Compared to block design experi-
ments, event-related experiments are more versatile be-
cause they can capture the temporal profile of the re-
sponse, thereby potentially providing timing information.
In addition, the event-related approach is also capable of
capturing trial-to-trial variations that may arise from fac-
tors such as habituation, learning, arousal, and attention.
One obvious drawback of ER-fMRI is the loss of SNR due
to the transient nature of the event-related response and
the time needed between trials. This loss of SNR, however,
has not impeded advances in ER-fMRI. Among these ad-
vances are the application of ER-fMRI to various high-
order cognitive tasks, the investigation of the utility of
epoch-dependent information, and the adaptation of meth-
ods developed for ERP to fMRI. Work reported by Buckner
et al. (9) demonstrated that higher-order cognitive tasks
such as activation of the prefrontal cortex during word-
stem completion was possible despite the fact that such
paradigms typically elicit much smaller signal changes
than sensory stimulation paradigms. At high fields it was
demonstrated that SNR is sufficient for detecting activa-
tion responses to individual trials without epoch averag-
ing (10). Experiments with intermixed trial types such as
infrequent events (11) or randomized go/no-go paradigms
(12) have allowed the detection of responses to rare events,
demonstrating the flexibility of ER-fMRI.
Currently, most ER-fMRI studies are performed using an
approach that relies on inter-epoch averaging as first in-
troduced by Buckner et al. (9). Despite the success of this
approach, which is robust in the presence of a low SNR,
the averaging ignores the information associated with each
execution of the task and assumes that subject behavior
and brain function do not vary during repeated trials. In
fact, trial-to-trial variations are frequently what is of inter-
est and can be correlated with task/performance variables
to reveal relevant information. The possibility of exploring
these time-dependent modulations is the greatest benefit
ER studies have to offer. To date, however, true single trial
experiments have been reported only at high fields due to
SNR limitations (10). Therefore, an improvement in SNR is
greatly needed to fully exploit the potential of ER-fMRI,
and is thus the focus of the present work.
In developing techniques for ER-fMRI, methods intro-
duced for ERP can be adapted. An example of such an
adaptation is the work by Dale and Buckner (13) on the
method of selective averaging. In the present work, which
focuses on denoising, the method of time-varying Wiener
filtering (14,15), along with the subsequent extension to
the wavelet transform domain (16) has been adapted for
processing ER-fMRI data. This technique is mainly aimed
at improving the SNR in the activated time courses before
they are characterized for aspects such as onset, duration,
and amplitude of the BOLD response. In adapting these
approaches for processing ER-fMRI data, several modifica-
tions were made. The most important modification is that
in order to preserve the epoch-to-epoch variation, the filter
is applied to individual epoch data. In addition, filtering is
done under the framework of the translation-invariant
wavelet transform (also known as the stationary wavelet
transform), rather than the conventional discrete wavelet
transform, in order to enhance the denoising performance.
The main purpose of this work is to develop a denoising
technique for individual time courses that have already
746
University of Minnesota, Center for Magnetic Resonance Research, Minne-
apolis, Minnesota.
Grant sponsor: NIH; Grant numbers: RO1MH55346; RR08079; R03MH59245.
*Correspondence to: Xiaoping Hu, Ph.D., University of Minnesota Medical
School, Center for Magnetic Resonance Research, 2021 6th Street, S.E.,
Minneapolis, MN 55455. E-mail: xiaoping@cmrr.umn.edu
Received 19 July 1999; revised 18 April 2000; accepted 23 June 2000.
Magnetic Resonance in Medicine 44:746–757 (2000)
© 2000 Wiley-Liss, Inc.

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been identified as “active” through some other method
such as correlation analysis. Since the event-related design
can provide the temporal information regarding the brain’s
response to events, enhancing the SNR of individual
evoked responses will increase the accuracy of their tem-
poral characterization, thereby improving the quantifica-
tion of the commonalities and differences that may occur
across repeated trials. In addition, using the filtering
method as a preprocessing step to model-free algorithms,
such as data clustering (17–19), may lead to improved
activation detection. It should be noted that a model-free
approach is more appropriate in this situation since the
noise structure of the filtered data is complicated, making
statistical analysis using conventional methods difficult.
THEORY AND ALGORITHM
In this section the concepts of Wiener filtering and its
time-variant version are first briefly reviewed. Subse-
quently, a description of the discrete wavelet transform
along with its stationary version is given. Finally, the
Wiener filter using the stationary wavelet transform is
described.
Wiener Filter
The Wiener filter is the solution to the linear minimum
square error problem of estimating a signal, x[n], from a
measurement consisting of signal plus noise, y[n] ? x[n] ?
v[n], using a filter, h[n]. This estimation can be expressed
as:
x ˆ?n? ? h?n?*y?n?, [1]
which is the convolution of the measured signal with a
time invariant filter. Here h[n], determined by minimizing
the estimation error, defines the Wiener filter. It can be
shown (20) that the frequency response of the noncausal
Wiener filter is:
H??? ?
Px???
Px??? ? Pv???
[2]
where Px(?) and Pv(?) represent the power spectral density
of the true signal and the noise, respectively. Intuitively,
the action of Eq. [2] is to keep frequency bands where the
signal power is much stronger than that of the noise and to
severely attenuate frequencies where noise predominates.
In deriving the time-invariant Wiener filter, both x and v
are assumed to be stationary. For many applications, the
assumption of stationary signal and noise characteristics is
not completely satisfied. A natural extension of the Wiener
filter is to allow it to be time-varying. In principle, this
should only require substituting the signal and noise
power spectral densities with their time-dependent coun-
terparts. In the context of ERP analysis, de Weerd and Kap
(14,15) introduced and implemented a time-varying Wie-
ner filter. An alternative is to perform the filtering in the
wavelet domain, where localization is achieved in both the
frequency and time domains (16,21).
Stationary Wavelet Transform
The present work relies on the stationary wavelet trans-
form (SWT) (22–24), which has the advantage of being
shift-invariant, albeit computationally more expensive
than standard implementations of the discrete wavelet
transform (DWT). Detailed description of the SWT can be
found elsewhere (23). For completeness, its essence is
summarized in this subsection.
Since the definition of the SWT follows from that of the
DWT, the definition of the DWT is first described. Given a
time series, T ? {x[0], x[1], . . . , x[N ? 1]} with N ? 2Jfor
some integer J, the DWT is performed based on the appli-
cation of two filters: a low pass filter H and a high pass
filter G, both in conjunction with a binary decimation
operator D0. The filter H is defined by a sequence {hn}n?Z,
wherehn’s satisfythe orthogonality
hnhn?2j? 0 for any integer j. The filter G is defined as
{gn}n?Zwith gn? (?1)nh1?n. Applying H on the time series
T results in (Hx)k? ?nhn?kx[n], while applying D0on T
produces (D0x)j? x[2j]. Now, define the smooth compo-
nent of the given time series at resolution level J as the
original time series itself:
condition
?n
sJ? ?sJ?n??, with sJ?n? ? x?n?
for n ? 0, 1, . . . , 2J? 1 [3]
and the smooth and the detail components of the time
series at resolution level j as:
sj? D0Hsj?1and dj? D0Gsj?1
[4]
where sjand djare sequences of 2jterms. The DWT of the
time series from level J to some level L ? J is given by 2J
terms:
Q ? ?dJ?1, dJ?2, . . . , dL, sL?
[5]
which provide a complete representation of the given time
series.
It can be shown (23,24) that the binary decimation op-
erator D0causes shift-variance and hence leads to prob-
lems concerning stability. To remedy the situation, the
high and low pass filters are applied to the given time
series without decimation in the SWT. To describe this
mathematically, a new operator Z is defined such that
(Zx)2j? x[j] and (Zx)2j?1? 0, and H[r]and G[r]are given
by {Zrh} and {Zrg}, where Zrdenotes the sequential appli-
cation of the operator Z r times. To perform the SWT, set:
sJ?n? ? x?n? for n ? 0, 1, . . . , 2J? 1 [6]
and define the smooth and detail components of the time
series at level j as:
sj?1? H?J?j?sjand dj?1? G?J?j?sj
[7]
where sj?1and dj?1are now sequences of 2Jterms. The
resultant (J ? L)2Jterms:
Z ? ?dJ?1, dJ?2, . . . , dL, sL?
[8]
Wavelet Transform-Based Wiener Filtering747

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give the stationary wavelet transform, a redundant repre-
sentation of the original time series.
Wavelet-Based Wiener Filter
Roughly speaking, the denoising filter to be derived here
and employed throughout the rest of this article involves a
modification of Eq. [2] expressed in the wavelet domain.
The concept of a wavelet-based Wiener filter is, of course,
not new. As mentioned previously, similar ideas have
been explored and applied to ERP (16) and electrocardio-
gram (ECG) data (21). The approach taken here is different
from the previous approaches in two aspects. First, instead
of designing a filter to denoise an averaged epoch, the filter
implemented here is intended for denoising individual
epochs. Second, the filter is applied in the SWT domain
rather than the DWT domain. The first aspect is needed
because epoch-to-epoch variations may be of interest in
ER-fMRI, as discussed earlier. When the activation signal
varies slightly from one epoch to another, it is advanta-
geous to denoise each epoch separately so that the pro-
cessed data still reflects this epoch-to-epoch variation. The
second aspect of using the SWT is essential because the
redundancy provided by this method leads to a more sta-
ble filter (24).
The derivation of Eq. [2] assumes that the power densi-
ties of the signal and noise are known. In reality, they are
to be estimated from the measured data. To this end, a
number of approaches have been suggested. For example,
de Weerd and Kap (15) introduced a method for processing
ERP data based on two types of averages, along with the
assumption that each measured evoked response consists
of a deterministic, time-varying component plus a random
noise component. In this work, a method similar to that
introduced by Bertrand et al. (16) is used.
Let us denote a time course with K epochs, each epoch
consisting of N samples, as:
?x0?0?, . . . , x0?N ? 1?, . . . , xk?0?, . . . ,
xk?N ? 1?, . . . , xK?1?0?, . . . , xK?1?N ? 1??. [9]
Hence, the ensemble average reads:
x ˜?n? ?1
K?
k?0
K?1
xk?n?. [10]
Performing the SWT on an individual epoch k yields:
?dk
J?1?0?, . . . , dk
J?1?N ? 1?, . . . , dk
J?2?0?, . . . ,
dk
J?2?N ? 1?, . . . , dk
L?0?, . . . ,
dk
L?N ? 1?, sk
L?0?, . . . , sk
L?N ? 1??.[11]
Similarly, the same operation on ensemble average yields:
?d˜J?1?0?, . . . , d˜J?1?N ? 1?, . . . , d˜J?2?0?, . . . ,
d˜J?2?N ? 1?, . . . , d˜L?0?, . . . , d˜L?N ? 1?,
s ˜L?0?, . . . , s ˜L?N ? 1??
[12]
In the wavelet domain, the desired Wiener filter takes the
form:
H?j, n? ?
Px?j, n?
Px?j, n? ? Pv?j, n?
[13]
where Px(j,n) is the power density corresponding to the
detailed component of true signal x in location n at reso-
lution level j. Pv(j,n) is the corresponding term of the noise.
It can be shown that the denominator can be approximated
by:
Px?j, n? ? Pv?j, n? ?1
K?
k?0
K?1
?dk
j?n??2
[14]
with the approximation approaching equality as K tends to
?. To get an estimate of the numerator of Eq. [13], it is
noted that:
?d˜j?n??2? Px?j, n? ?1
KPv?j, n?.[15]
This progression from Eq. [13] follows very closely the
results in (16). Combining Eqs. [14, 15] gives:
Px?j, n? ?
K
K ? 1?d˜j?n??2
??
1
K?
1
K ? 1? ?
k?0
K?1
?dk
j?n??2.[16]
By substituting Eqs. [14] and [16] into Eq. [13], we obtain:
H?j, n? ??
K
K ? 1?d˜j?n??2??
1
K?
1
K ? 1? ?
k?0
K?1
?dk
j?n??2
1
K?
k?0
K?1
?dk
j?n??2
?
.
[17]
Since it is only possible to obtain approximate values for
the power densities in Eq. [13], the values of H(j,n) from
Eq. [17] are not constrained to the range of [0,1]. Thus,
H(j,n) is truncated so that its values must fall within this
desired range; negative values are replaced with zeros, and
values greater than one are replaced with ones. Now, the
denoised experimental data in the wavelet domain can be
written as:
?dˆ
k
J?1?0?, . . . , dˆ
k
J?1?N ? 1?, . . . , dˆ
k
j?0?, . . . ,
dˆ
k
j?N ? 1?, . . . , dˆ
k
L?0?, . . . , dˆ
k
L?N ? 1?,
sk
L?0?, . . . , sk
L?N ? 1??
[18]
where dˆk
to the smooth components of each epoch, sk
j[n]?dk
j[n]H(j,n). Note that filtering is not applied
L[1], . . , sk
L[N].
748LaConte et al.

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In view of Eq. [13], the action by Eq. [18] on the raw data
is to shrink each of its wavelet coefficients according to the
ratio between the power density due to noise and the
power density due to signal plus noise in the correspond-
ing component.
It should be noted that even though an epoch-to-epoch
variation is present in the data, the estimation of the Wie-
ner filter assumes that this variation is negligible. There-
fore, the estimated filter represents an “average” Wiener
filter and is not necessarily optimal for each epoch from a
theoretical point of view (see Discussion).
METHODS
The above algorithm has been implemented using the
WaveLab package (25). All supporting routines were writ-
ten in Matlab (MathWorks, Natick, MA). This approach
was investigated by applying it to simulated fMRI datasets
as well as activated time courses from experimental data.
In the simulation studies, the performance of the present
approach was measured by the reduction of RMS error and
by the improvement in detectability when it is used as a
preprocessing step for a model-free detection approach.
Further, these performance measures of the present ap-
proach were compared to those of an analogous Fourier
domain filter. In the experimental studies, the present
approach was applied to activated time courses from ex-
perimental data obtained with a visually guided motor
paradigm to illustrate its practical utility. All results re-
ported utilized a Daubechies filter of length 4, as imple-
mented in the WaveLab package.
Denoising Activated Pixels in Simulated Data
Simulated activation was generated using a signal of the
form (see Fig. 1a for an example):
x?t? ??1 ? exp?
?t
T1??
3
? exp?
?t
T2?
[19]
where T1and T2are constants that can be adjusted to
obtain the desired shape, and t represents the sampling
times (i.e., the image number within an epoch). In most of
the simulations described below, T1and T2took the values
of 5.0 and 7.5, respectively. This signal was replicated for
each epoch and added to either simulated Gaussian white
noise or experimentally acquired baseline data. The base-
line data were collected on a healthy adult male volunteer
using an EPI sequence (TE/TR: 25/500 ms, FOV: 20 ?
20 cm2, matrix: 64 ? 64, slice thickness: 5 mm,
2000 images) on a Siemens 1.5 T clinical scanner. To
account for possible drift in the data, the mean of each
epoch was removed from the corresponding epoch before
denoising.
Simulations were performed to characterize the denois-
ing algorithm with respect to SNR, the number of epochs,
and small epoch-to-epoch variations of the fMRI response.
These factors were examined to see how the performance
depends on the accuracy of power density estimation. In
these simulations, the root-mean-square (RMS) error, cal-
culated between the filtered time course and the true sig-
nal, is used as a performance measure and compared to the
RMS error in the original data. Each simulation was re-
peated 10 times. The mean and the standard deviation of
the results from the ten repetitions are reported. Finally,
simulations were performed to compare the wavelet-based
Wiener filter to the Fourier-based Wiener filter in order to
demonstrate the advantage of operating under the wavelet
domain (26,27).
SNR
Simulations to explore filter performance with varying
levels of SNR were performed. In this case, the synthetic
data were generated with both simulated Gaussian white
noise and baseline data. The SNR, defined as the standard
deviation of the simulated signal over the standard devia-
tion of the added noise, ranged from 0.25–4.0. The number
of epochs was eight, and the length of each epoch was
64 time samples.
Number of Epochs
The filter performance with respect to varying number of
epochs was studied. The SNR of the simulated data was
1.0 and baseline data were added to the simulated activity.
The number of epochs ranged from two to eight, with each
epoch having 64 time samples.
Epoch-to-Epoch Variations
The robustness of the filter was tested under the two con-
ditions of epoch-to-epoch variation in simulated time
courses. Two possible scenarios were investigated. The
first addresses variations in the amplitude of the response,
which may arise, for example, when a task of varying
difficulty is used. The second focuses on variations in the
duration of the response. For the condition of variable
amplitude, each epoch was generated as described above,
with the signal component (Eq. [19]) multiplied by a ran-
dom value ranging from 1.0–2.0. In the case of variable
response duration, the signal component of each epoch
was convolved with a boxcar function having a width
varying from 12–25 time samples. In both cases, the SNR
of the simulated data was 1.0 with baseline data as noise,
the number of epochs was eight, and the length of each
epoch was 64 time samples.
Fourier Domain Filtering
A comparison of the noncausal Wiener filter as described
by Eq. [2] with the SWT method was performed. The filter
coefficients were estimated as described by Eq. [17] with
the exchange of the wavelet coefficients with their Fourier
domain counterparts. The data generated to investigate the
performance of the SWT-based Wiener filter at different
SNR levels were also used for this comparison. Quantita-
tive comparison of the two methods was achieved by tak-
ing the difference of the average RMS error of the two
methods and dividing the result by the average RMS error
of the wavelet-filtered time series.
ROC Analysis
To study the impact of this method on detection of acti-
vation, ROC analysis was performed using simulated data
Wavelet Transform-Based Wiener Filtering 749

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in a manner similar to that described by Xiong et al. (28).
Four epochs of artificial activation (each with a duration of
64 images) with the signal pattern of a half-sinusoid and a
preassigned contrast level (ranging from 1–3%) were su-
perimposed on the experimental dataset. The spatial loca-
tions of the activated pixels correspond to a 5 ? 5 grid of
squares. Finally, an ROI encompassing the extent of the
brain was delineated and the pixels within the ROI were
FIG. 1. Simulated time courses before and after denoising. The horizontal axis represents image number and the vertical axis corresponds
to normalized signal intensity. a: The uncorrupted (true) signal. Baseline data were added to a to yield time courses with SNR of 0.5 (b1),
1.0 (c1), and 2.0 (d1), respectively. Their denoised counterparts are shown in b2, c2, and d2.
750LaConte et al.