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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008

Performance of Prewhitening Beamformingin MEG Dual Experimental Conditions
Kensuke Sekihara*, Kenneth E. Hild, II, Senior Member, IEEE, Sarang S. Dalal, andSrikantan S. Nagarajan, Senior Member, IEEE

Abstract—This paper presents an analysis on the performanceof the prewhitening beamformer when applied to magnetoencephalography (MEG) experiments involving dual (task andcontrol) conditions. We first analyze the method’s robustness to
two types of violations of the prerequisites for the prewhitening
method that may arise in real-life two-condition experiments.
In one type of violation, some sources exist only in the control
condition but not in the task condition. In the other type of violation, some signal sources exist both in the control and the task
conditions, and that they change intensity between the two conditions. Our analysis shows that the prewhitening method is very
robust to these nonideal conditions. In this paper, we also presenta theoretical analysis showing that the prewhitening method is
considerably insensitive to overestimation of the signal-subspacedimensionality. Therefore, the prewhitening beamformer does
not require accurate estimation of the signal subspace dimension.
Results of our theoretical analyses are validated in numerical experiments and in experiments using a real MEG data set obtained
during self-paced hand movements.
Index Terms—Adaptive beamforming, brain noise, interference
removal, magnetoencephalography (MEG), neuromagnetic sourcereconstruction, prewhitening.

I. INTRODUCTION
NE MAJOR problem with magnetoencephalography(MEG) measurements is that the measured MEG datacontain not only signals from brain regions of interest, but also
large interfering magnetic fields generated from spontaneous
brain activities all over the brain. Such background interference
degrades the quality of source reconstruction results, and often
makes interpreting the results difficult. Such background interference is sometimes referred to as brain noise or physiologicalnoise.
A common strategy for extracting the signal of interest frommeasurements overlapped with a large amount of interference
is to design experiments with dual (control and task) conditions. Subtraction between the reconstruction results obtainedO

Manuscript received December 25, 2006.
This work was supported in part by Grants-in-Aid from the Ministry of
Education, Science, Culture and Sports in Japan under Grant C16500296 to
K. Sekihara and supported in part by grants from the Whitaker Foundationand the National Institute of Health (R01-DC004855-01A1 and NS006435)
to S. Nagarajan. Asterisk indicates corresponding author.
*K. Sekihara is with the Department of Systems Design and Engineering,Tokyo Metropolitan University, Asahigaoka 6-6, Hino, Tokyo 191-0065, Japan.
K. E. Hild, S. D. Dalal, and S. S. Nagarajan are with the Department of Radiology, University of California, San Francisco, CA 94143 USA.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TBME.2008.915726

under these two conditions is a common procedure to reconstruct signal sources of interest [1]. (This subtraction is often
performed as a part of calculating pseudo- statistics, which isused for statistically evaluating the source configuration difference between the two conditions.) However, when the sourcereconstruction is performed with adaptive spatial filter methods[2]–[4], such subtraction-based methods cannot effectively remove the influence of the background interference [5]. This isbecause the influence of the background activity is not simply
additive. It involves spatial blur and source location bias, as will
be shown in our computer simulation in Section V.
Recently, we have proposed a novel prewhitening method
suitable for reconstructing sources from evoked measurements
overlapped with large background interference [6]. The goal ofthis paper is to show that the prewhitening method can also be
effective for measurements involving task and control conditions. We first analyze the prewhitening method’s robustness to
two types of violations of the prerequisites for the prewhitening
method that may arise in real-life two-condition experiments.
We refer to these two types violations as the control-only source
scenario and the modulating source scenario in this paper. In thecontrol-only-source scenario, some sources appear only in thecontrol measurements and that they do not appear in the task
measurements. In the modulating-source scenario, some signal
sources of interest exist both in the control and the task measurements, and they change intensity between the two conditions. In real-life measurements, one or both of these scenarios
may arise. In this paper we demonstrate that the prewhitening
method is still effective under these scenarios.
This paper also presents an analysis on the influence causedby the overestimation of the signal subspace dimensionality,
and it shows that the method is considerably insensitive to suchoverestimation. Therefore, the accurate determination of thesignal subspace dimension is not essential for implementing
the prewhitened beamformer, and an intentionally large value
can be used for the signal subspace dimension.
Section II briefly reviews the prewhitening method that has
already been proposed in [6]. Section III presents our theoreticalanalysis on the method’s performance under two realistic scenarios with dual-condition experiments. Section IV shows theeffects of overestimation of the signal subspace dimension. The
arguments in Sections III–IV are validated first by numericalexperiments in Section V and by an application to real MEG
data collected during self-paced finger flection in Section VI.
Throughout this paper, plain italics indicate scalars, lower-caseboldface italics indicate vectors, and upper-case boldface italics
indicate matrices. The eigenvalues are numbered in decreasingorder.

0018-9294/$25.00 © 2008 IEEE

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II. PREWHITENING BEAMFORMING UNDER
AN IDEAL SCENARIO
We use a model for measurements

holds. To estimate the signal covariance matrix, we first calculate the prewhitened measurement covariance matrix , which. Thus, from (4), we have theis defined as
relationshipexpressed as
(1)

(5)
where
is the magnetic field generated by signal sources
is the magnetic field generated by backgroundof interest,is the additive sensor noise. These vectors
activity, andcolumn vectors whereis the number of senare
sors. The covariance matrix of the measurements is denoted
such that
where
indicates an expectation
operator. We define the covariance matrix of the signal mag, such that
. We assume
netic field asthat the signal is low rank, i.e., the rank ofis , which issmaller than , the number of sensors. We also define the in, such that
terference-plus-sensor-noise covariance matrix. Note that thisis a positive definite matrix, because we assume that the sensor
noise is the white Gaussian noise uncorrelated between different
channels. Under the assumption that the signal source activity
is uncorrelated with the background interferences and sensor
noise, the relationship(2)
holds. In the conventional (nonprewhitened) minimum-variancespatial filter, the source power reconstruction
is obtainedusing this covariance matrix such that
(3)
where
is an
column vector representing the sensor
lead field in the estimated source direction1 at location . Equation (3) indicates that the influence of the additive interference is
not simply additive, but it affects the final source-reconstruction
in a highly nonlinear manner. The influresults throughence actually involves spatial blur and source location bias, as
will be shown in our computer simulation in Section V.
We next explain the prewhitening method of estimating the; the method was first reported insignal covariance matrix[6]. We here assume the ideal scenario, in which control-state
contain only the interference and sensor
measurements
noise, i.e.,

and the interference-plus-noise covariance matrix
canbe obtained from such control-state measurements. Namely,
defining the covariance matrix from the control measurements, the relationshipas , i.e.,Control:
Task:

(4)

1The method of estimating the source orientation at each voxel location ispresented in [7] and [8].

where
(6)
We define the eigenvalues and eigenvectors of an
matrixasand. Sinceis a positive semidefinite matrix with rank andis a nonsinis also a positive semidefinite matrix with rank
gular matrix,. Thus, the eigenvalues
are positive and the otherare zero. (Here, we assume that the
eigenvalues
eigenvalues are ordered in a decreasing manner.) Namely, wehave(7)Therefore, the eigendecomposition of

can be expressed as(8)

The above equation indicates that the largest eigenvalues ofare greater than 1 and the corresponding eigenvectors are the. The eigensame as those for the nonzero eigenvalues ofgreater than 1 are referred to as the signal-levelvalues of
eigenvalues and their corresponding eigenvectors as the signallevel eigenvectors. Equation (8) indicates that it is possible tofrom the signal-level eigenvalues and their correretrieveassponding eigenvectors of . That is, defining a matrix, the signal covariance matrix,, can beobtained using(9)andare unIn actual cases, the covariance matrices
known, and instead we should use the sample covariance matrices, which are obtained using(10)
are the time points contained in a certain time
where
. Using (9),window. We define such thatcan be obtained
the estimate of the signal covariance matrix
using
(11)

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where

, and

are the signal-level

eigenvectors of . Given the estimate of the signal covariancematrix , a reasonable estimate of the signal-plus-sensor-noise, can be obtained using
covariance matrix,

In deriving (15), we assume that the activity of control-only
sources is uncorrelated with the interference and sensor noise.
Using (15), we have
(17)

(12)
where is the regularization constant that should be set close
to the variance of the sensor noise . (Actually, since the noise
variance is unknown, an appropriate value of should be determined from the measured data, and some empirical methods,
such as that of using the minimum-eigenvalue of , are employed to determine .) Consequently, using the minimum-variance spatial filter, the prewhitening source power reconstruction
free from the influence of the background activity, can be obtained using

and thus(18)
where. Sinceis a nonnegative defiis a nonsingular matrix,
nite matrix with rank andis a nonnegative definite matrix with rank . Thus,
is decomposed as

(19)

(13)

are nonzero eigenvalues of,where ,
and are the corresponding eigenvectors. Substituting (7) and(19) into (18), we haveIII. PREWHITENING BEAMFORMING UNDERNONIDEAL SCENARIOSIn this section, we analyze the performance of the
prewhitening method under two kinds of nonideal scenariosthat may arise in real-life task and control-type measurements.
In the scenario argued first, there are some sources that appear
only in the control state and do not appear in the task state. Suchsources are called the control-only sources in this paper, andthis scenario is referred to as the control-only-source scenario.
In the scenario argued next, the signal sources of interest are
active in the control state as well as in the task state but theychange their intensities between the two states. Such sources arecalled the modulating sources in this paper, and this scenario isreferred to as the modulating-source scenario.

(20)
When the control-only sources are well separated from thesignal sources of interest, the relationship(21)
approximately holds. Under this assumption, we will show thata set of vectors(22)

A. Control-Only Source Scenario
In the control-only-source scenario, we assume that there aresources that exist only in the control state and do not appear in
the task state. We also assume that the observed signal space is. When such control-only sources
still low-rank, i.e.,
can be expressed asexist, the control state measurements

(14)
where
indicates the magnetic field generated by controlonly sources. The covariance matrix relationships are then expressed asControl:Task:

are the eigenvectors of , where
orthonormal basis set of the subspace

are the, which is defined as.

We first show that the relationship(23)
holds. That is, we show that the vectors (where)
are the eigenvectors of and their corresponding eigenvalues. To show this, we calculate the right multiplicationareto , and using (20) this multiplicationofresults in

(15)

where

(16)

(24)

SEKIHARA et al.: PERFORMANCE OF PREWHITENING BEAMFORMING

The first term in the right-hand side becomes
term becomes , because the relationship1115. The second

are the eigenvectors ofdecreasing order, are. The corresponding eigenvalues, in a
(30)

(25)
holds. The validity of (25) is shown in the Appendix. The third
term in the right-hand side of (24) vanishes due to the orthogonality assumption in (21). Therefore, we can derive the rela(wheretionship in (23), i.e., we can show that the vectors) are the eigenvectors of .
Next, we show that the relationshipare greater
Here, the largest eigenvalues
than 1, and therefore, (9) is still effective at retrieving , evenwhen the control-only sources exist.
In general, however, the subspace angle between
andmay not be so large
and the assumption that these two subspaces are orthogonal
may not be satisfied. In such cases, (23) is changed to

(31)
(26)
holds. That is, we show that the vectors
are the eigenvectors of and the corresponding eigenvalues
are equal to 1. To show this relationship, we calculate the rightmultiplication of to , which is equal toThis equation shows thatis no longer the
eigenvector of and the second term on the right-hand side of(31) indicates the error term. Thus, if the relationship

(32)

(27)
is orthogonal to both the subspace spanned byand that spanned by, the only
nonzero term in the right-hand side is the second term, which. Thus, we
is equal to , because
have proved (26).
Finally, we calculate the right multiplication of
to , which produces that

holds, the error term is negligibly smaller than the first term,
are still approximately the signal-level eigenvectors ofand. Conversely, when the error term is not small, the signal-covariance estimate obtained from (9) could be erroneous.

Since

B. Modulating-Source Scenario
We next examine the modulating-source scenario. We define
the covariance matrix of the signal activity in the control state
as . Then we have the relationship
Control:
Task:

(33)

Thus, we have
(34)

(28)
On the right-hand side, the first term becomes zero due to theorthogonality assumption in (21) and the second term becomeszero due to the orthogonality relationship between the signal andthe noise subspaces. Thus, we have(29)
are eigenvectors ofand the
Therefore,. We can also show that
corresponding eigenvalues arethese eigenvalues are positive but less than 1, i.e.,, although we do not include the proof here. In summary, we
have shown that the vectors

We consider a general case where some signal sources have theirintensities greater in the control state than in the task state, butothers have their intensities smaller in the control state than in
the task state. The power of the th signal source in the task andand.
the control conditions are denoted, respectively,
have
We assume that the signal sources withthe relationship of, and that the signal sources withhave the relationship. Therefore,
defining, we have
(35)

where
(36)

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(37)
Therefore, we have(38)
and thus
(39)
where

Because bothandare the positive semidefinite matrices,(39) is in principle the same as (18), and exactly the same arguments hold as those in Section III-A. Therefore, we can estimate
by using

(40)
where

is a matrix containing

tors of . We can estimate
by changing the role ofsuch thatThat is, we first calculate
using
and then obtain an estimate ofaccurately determine because the eigenvalue spectrum does
not have a clear separation between these two subspaces. Here,
we presents an analysis on the influence caused by the overestimation of the signal subspace dimensionality, and we showthat the prewhitening method is very insensitive to such overis not
estimation. Therefore, the accurate determination ofneeded and we can use an intentionally large to implement
the prewhitening method.
In the following, we discuss the influence caused when theis overestimated. We
dimension of the signal-subspace ofassume the ideal scenario in this section. Let us assume thatthe signal subspace dimension is overestimated as
and defineas. Ideally, theprewhitened data covariance matrix, , hassignal-level
eigenvalues equal toeigenvalues greater than 1 anddoes
1. Therefore, according to (9), the overestimation of
because the relanot affect the signal covariance estimate
holds. However, the prewhitened
tionshipis usually estimated from finite timecovariance matrix
samples, and in such cases, the noise-level eigenvalues are
generally not equal to 1. We denote the noise-level eigenvalues
asof the estimated prewhitened covariance matrix
and assume
for. Then, the estimatedsignal covariance matrix, , is expressed in this case as

eigenvecand

.
,

(44)
(41)
In the equation above,is defined as(wherewhich is a matrix containing,) signal-level eigenvectors of . The prewhitening method in which theare reversed is referred to as the “flipped”
roles of andprewhitening method in this paper. Therefore, the sources thatare stronger in the task state than in the control state can bereconstructed using(42)
The sources that are stronger in the control state than in the taskstate can be reconstructed using the flipped prewhitening, i.e.,

The second term on the right-hand side of the above equation
indicates the error term caused by the overestimation. The error
term, , is expressed as
(45)

where
.
Let us define the signal and noise subspaces of the measureand, i.e.,
ment covariance matrix , as

where

is the th eigenvector of . Then, assuming that
, we can expand
using such that

(43)
(46)
IV. OVERESTIMATION OF SIGNAL-SUBSPACE DIMENSIONALITY
The prewhitening method requires us to determine , the di). Thismension of the signal subspace of (or the rank ofdetermination is usually performed by counting the number ofdistinctively large eigenvalues of data covariance matrix . In
actual MEG measurements, however, it is often problematic to. On the rightwhere
hand side of (46), the first term is the signal subspace component and the second term is the noise subspace component. To
obtain the source reconstruction in (13), we need to calculate
the signal-plus-sensor-noise covariance matrix. The estimate of

SEKIHARA et al.: PERFORMANCE OF PREWHITENING BEAMFORMING

the signal-plus-sensor-noise covariance matrix,
case given by1117, is in this(47)
where is the th eigenvalue of , and we assume thatfor.
The equation above indicates that the influence of the overestimation is mainly an increase of the regularization constant.
The regularization in the minimum variance beamformer,
known as diagonal loading, has been widely used and a significant increase of the regularization constant is known to causea spatial blur in the reconstruction results [5]. Therefore, theoverestimation of the signal subspace dimension should cause aand thespatial blur. However, the blur should not be large if
are small. In Section V, examples are presented
resultantin which the prewhitening method can still provide successfulreconstruction, even when the signal subspace dimension issignificantly overestimated. In the analysis presented in thissection, we assume the ideal scenario. Considering the fact thatlarge eigenvalues are the same in the ideal and the nonideal
scenarios, it is obvious that the arguments here are also valid
for the nonideal scenarios.
V. COMPUTER SIMULATION
A. Data Generation
A computer simulation was performed to demonstrate thevalidity of the arguments in the preceding sections. In our
experiments, we used a sensor alignment of the 275-sensor
array from the Omega™ (VMS Medtech, Coquitlam, Canada)
neuromagnetometer. Three sources were assumed to exist on
cm , and theircoordinates werea single plane(-2.1,9.5) cm, (2.6,10.5) cm, and (1.4,7.5) cm, respectively.
The source-sensor configuration and the coordinate system aredepicted in Fig. 1(a). The spherical homogeneous conductor
model [10] with the sphere origin set at (0,0,4) cm was used
for the forward calculation. The powers of the three sourceswere set equal in the sensor-domain, i.e., the relationship,, held where , ,were the location, the power, and the lead field vector
andof the th source, respectively. Multiepoch measurements were
simulated.
Each epoch had two sets of data: the task and the control data sets. The task and the control data sets in theth epoch are denotedand, respectively, andthey are expressed as, and, whereis the number oftime points. Since
was set at 600 in our computer simulation,andresulted in 275 600 spatio-temporaland, we assumed
data matrices. To calculate
uncorrelated sinusoidal time courses for the three sources; thesource time course for the th source and for the th epoch isexpressed as,
where
is the time window equal to, and the constantcontrols the frequency, which was set at 6.3 for the first

Fig. 1. (a) Source-sensor configuration and the coordinate system used in the
numerical experiments. The plane at x = 0 cm is shown. The small filled
circles indicate the locations of the three sources. The large circle indicatesthe boundary of the sphere used for the forward calculations. The center ofthe sphere was set to (0,0,4) cm. (b) Representative examples of the generated
single-epoch data B (upper panel), and B (lower panel).

source, 9.1 for the second source, and 13.1 for the third source.
The phase offset
was generated using the random number
between 0 and , and different random number was used for
depending on , , and the two conditions. We therefore
simulated induced source activities, which are elicited by thestimulus but not phase-locked to it.
B. Simulation for the Control-Only-Source Scenario
We first check the performance of the prewhitening method
under the existence of a control-only source. In this computer
simulation, the first source is the control-only source, and thesecond and the third sources are the signal sources, which apcontains the magnetic
pear only in the task state. Namely,
field from the second and the third sources, andcontains
the magnetic field from the first source. Real spontaneous MEG, and the signal-to-interferwas used as the interference
ence ratio (SIR) was set equal to 0.3. That is, values of the source
were determined in order for the SIR
power
defined as
to 0.3. A total of 96 epochs of

to be equal
and

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Fig. 2. Results of the source reconstruction experiments simulating the control-only-source scenario. (a) Results of conventional reconstruction for the task
condition, 8(r ). (b) Results of conventional reconstruction for the control condition, 8 (r ). (c) Results of the prewhitening source reconstruction. (d) Resultsof the flipped prewhitening source reconstruction.

were generated. The representative examples ofandare shown in Fig. 1(b). The sample covariance matrices
andwere calculated using
and
where
indicates thenumber of epochs, which is equal to 96 in this computer simulation. The total covariance matrix was calculated for later use,
using
(48)
The conventional beamformer source reconstruction was
performed for the control and the task data, using
and
where
and
are the source power reconstruction forthe control and the task states, respectively. The results areshown in Fig. 2(a) and (b). These results contain severe blurdue to the interference inherent in the task and the control
data. The source reconstruction was next performed withthe prewhitening estimate of the signal covariance matrix, using (13). The prewhitening estimate of,, wasalso obtained from the flipped prewhitening method using, and the source reconstruction was
replaced by. The results ofperformed using (13) withprewhitening and flipped prewhitening source reconstructions
are shown in Fig. 2(c) and (d). Compared to the results in
Fig. 2(a) and (b), the results in Fig. 2(c) and (d) show thatthe signal and the control-only sources were reconstructedat the correct locations with much higher spatial resolution,
demonstrating that the prewhitening method is still effective inthe control-only-source scenario.
C. Simulation for the Modulating-Source Scenario
In this computer simulation, the intensity of the first source
was decreased by 30% from the control to the task conditions,Fig. 3. Results of the source reconstruction experiments simulating the modulating-source scenario. (a) Results of conventional reconstruction for the task
condition, 8(r ). (b) Results of conventional reconstruction for the control condition, 8 (r ). (c) Results of calculating the pseudo-F image F (r ). (d) Resultsof applying the prewhitening source reconstruction. The signal subspace dimension was set to 15. (e) Results of the flipped prewhitening source reconstruction.

while the intensity of the third source was increased by 30%
from the control to the task conditions. The intensity of thesecond source remained the same between the two conditions.
The SIR was set equal to 0.3.
The existing method of processing this type of data
calculates the following pseudo-F image[3], such that
, whereandare obtained using
, and
, and
where the beamformer weight is obtained using(49)
is calIn the above expression, the total covariance matrix,, andculated using (48). The results of calculating
are shown in Fig. 3(a)–(c), respectively. In the pseudo-F
image in (c), the third source forms a positive peak, and thefirst source forms a negative peak. Although these peaks areblurred and the peak locations are biased, the pseudo-F image
can at least detect these two sources. Next, the methodsof prewhitening and flipped prewhitening source reconstruction
were performed, and the results are shown in Fig. 3(d) and (e).
In these results, the signal source and the control-only source
form clear peaks around the correct locations of these sources,
although a small bias of the reconstructed source location canbe observed, particularly for the third source.
To compare the source localization biases for the pseudo-F
and the prewhitening results, we performed a Monte Carlo-type

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Fig. 5. (a) Eigenvalue spectrum of R used for reconstruction in Fig. 3(d).
(b) The results of the prewhitening source reconstruction obtained with thesignal subspace dimension set to 30.

Fig. 4. Results of the Monte Carlo-type computer simulation performed to estimate the amount of the source localization bias. The mean of the 40 sets ofthe calculated distance between the true and the estimated locations of the thirdsource are plotted for four SIR values. The solid line represents the averagesource localization bias for the prewhitening beamforming, and the broken line
represents that for the pseudo-F method. The error bars indicate 1 standard
deviation.

6

computer simulation in which 40 data sets, each containing
and, were generated and 40 sets of reconstruction results were obtained. In each set of the results,
the distance between the peak location and the true locationof the third source was calculated as the source localization
bias. The mean and the standard deviation of the 40 sets of thesource-bias results are plotted for four SIR values in Fig. 4. (Thevalues of 1/SIR were set to 1, 2, 3, and 4.) According to theseresults, the amount of source bias is almost the same for the
prewhitening and pseudo-F results when the SIR is moderately
SIR. However, when the SIR is low
SIR
,
high
the prewhitening results have a significantly smaller source bias.
These results demonstrate the effectiveness and superiority ofthe prewhitening method in the modulating-source scenario.
Finally, we show the robustness of the prewhitening method
to the overestimation of the signal-subspace dimension . The
eigenvalue spectrum of for obtaining the results in Fig. 3(d)is shown in Fig. 5(a). This spectrum indicates that there is noclear separation between the noise- and the signal-level eigenvalues, and there should be some ambiguity when determining
the signal subspace dimension. When obtaining the results in
Fig. 3(d), the signal subspace dimension was set to 15. The reconstruction results obtained with the signal subspace dimension set to 30 are shown in Fig. 5(b). Results almost identicalto those in Fig. 3(d) were obtained in spite of the fact that thesignal subspace dimension was significantly overestimated inthis case.
VI. EXPERIMENTThe effectiveness of the prewhitening method is further
demonstrated using real MEG data collected during finger flection. The measurement was performed using the 275-channel

Omega-275™ whole-cortex biomagnetometer installed at the
Biomagnetic Imaging Laboratory, University of California, San
Francisco. Here, a subject was asked to press a button with hisright-index finger every 3–4 s. The onset of the movement was
indicated by a button press and defined as the time-origin. A
total of 200 epochs were acquired at a 1-kHz sampling rate.
We set two time windows for covariance matrix estimation:
the first from 1000–1300 ms, and the second from 300–0 ms.
The Fourier transforms of the th epoch data in the first andthe second time windows are respectively denoted
and. The frequency-domain sample covariance matrices forand , are obtained
the first and the second time windows,
using

(50)

where
and . In this equation, is the epoch index andthe notation
indicates the summation over a specificfrequency band, andwas set to the beta-band region
between 15 and 25 Hz in our experiments. In these experiments,
andwere used as the task and the control covariancematrices.
It is well-known that the intensity of the magnetic field in
the beta-band spectral region decreases preceding and during
the button-press finger movements [11]. Therefore, these measurements represent the modulating-source scenario where
signal sources are present both in the task and control conditions but modulated in their amplitudes. First, we calculated
the pseudo-F map
with the results shown in Fig. 6(a).
Although the pseudo-F map was able to detect source activities
in the left temporal region, the results are considerably blurry.
We next applied the prewhitening source reconstruction andthe results are shown in Fig. 6(b). We can see that the proposed
method can reconstruct a clear, localized source in the left
temporal region. The MRI overlay of these results is shown in
Fig. 6(c). The overlay shows that the center of the reconstructedactivity is located in contralateral hand-motor cortex. The

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prewhitening method are largely insensitive to the overestimation of the signal subspace dimension.
APPENDIX
This appendix presents the proof of (25). We first point out
that the following relationship holds:(51)
indicates the empty set. Although we do not
where
provide the formal proof, it is easy to prove this relationship. Because this relationship holds, we can then. That is,showbelongs to the complementary subspace of, which is equal to,. Therefore, becausei.e.,is the projector onto
, theapplication of this projector to results inwhich is (25).
REFERENCES

Fig. 6. Results of the source reconstruction experiment using hand-motor MEG
data. The maximum-intensity projections of (a) the pseudo-F results F (r ) and(b) the prewhitening results. The left, middle, and the right columns, respectively, show the maximum intensity projections of the 3-D reconstruction onto
the axial, coronal, and sagittal planes. The upper-case letters L and R show theleft and the right hemispheres. The signal subspace dimension was set to 15.
(c) MRI overlay of the prewhitening source reconstruction results in (b). (d) The
eigenvalue spectrum of R used for reconstructing the results in (b). (e) The results of the prewhitening source reconstruction obtained with the signal subspace dimension set to 25.

results in Fig. 6(b) and (c) demonstrate the effectiveness of the
prewhitening source reconstruction for this data set.
We again check the sensitivity of the prewhitening method tothe overestimation of the signal subspace dimension. The eigenvalue spectrum of is shown in Fig. 6(d). We can see that there
is no clear separation between the noise and the signal level
eigenvalues and that there would be some ambiguity when determining the signal-subspace dimension. When obtaining theresults in Fig. 6(b), the signal-subspace dimension was set to
15. (This value was determined by our computer algorithm that
detects a point where the spectrum starts to rise above the noise
slope.) The reconstruction results obtained with the dimension
set at 25 are shown in Fig. 6(e). These results are almost identical to those in Fig. 6(b), demonstrating that results from the[1] W. Chau, A. R. McIntosh, S. E. Robinson, M. Schulz, and C. Pantev,“Improving permutation test power for group analysis of spatially filtered MEG data,” NeuroImage, vol. 23, pp. 983–996, 2004.
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SEKIHARA et al.: PERFORMANCE OF PREWHITENING BEAMFORMING

Kensuke Sekihara received the M.S. and Ph.D. degrees from the Tokyo Institute of Technology, Tokyo,
Japan, in 1976 and 1987, respectively.
From 1976 to 2000, he worked with Central
Research Laboratory, Hitachi, Ltd., Tokyo. He wasa Visiting Research Scientist at Stanford University,
Stanford, CA from 1985 to 1986, and at Basic Development, Siemens Medical Engineering, Erlangen,Germany from 1991 to 1992. From 1996 to 2000, He
worked with “Mind Articulation,” a research projectsponsored by the Japan Science and Technology
Corporation. He is currently a Professor at the Department of Systems Design
and Engineering, Tokyo Metropolitan University, Tokyo. His research interests
include neuromagnetic source reconstruction and statistical signal processing,
especially its application to functional neuroimaging.
Dr. Sekihara is a member of IEEE Medicine and Biology Society and theIEEE Signal Processing Society.

Kenneth E. Hild, II (M’90–SM’05) received the
B.S. and M.Sc. degrees in electrical engineering, withemphasis in signal processing, communications, andcontrols, from the University of Oklahoma, Norman,
in 1992 and 1996, respectively, and the Ph.D. degree
in electrical engineering from the University of
Florida, Gainesville, in 2003, where he studied
information-theoretic learning and blind source
separation in the Computational NeuroEngineering
Laboratory. Dr. Hild has also studied biomedical
informatics at Stanford University, Palo Alto.
From 1995 to 1999 he was employed at Seagate Technologies, Inc., where
he served as an Advisory Development Engineer in the Advanced Concepts
group. From 2000 to 2003 he taught several graduate-level classes on adaptive filter theory and stochastic processes at the University of Florida. He iscurrently employed at the Biomagnetic Imaging Laboratory in the Departmentof Radiology, University of California at San Francisco, where he is applying
variational Bayesian techniques for biomedical signal processing of encephalographic and cardiographic data.
Dr. Hild is a member of Tau Beta Pi, Eta Kappa Nu, and the International
Society for Brain Electromagnetic Topography.

1121

Sarang S. Dalal was born in Hayward, CA, in1978. He received the B.S. degree in biomedical
engineering, with a concentration in electricalengineering and a minor in psychology, from The
Johns Hopkins University, Baltimore, MD, in 2000
and the Ph.D. degree in bioengineering, jointly with
the University of California, San Francisco (UCSF)
and the University of California, Berkeley in thelaboratory of Dr. Srikantan Nagarajan, in 2007.
His research interests include neuromagnetic
source reconstruction, time-frequency analyses,
cortical connectivity, and auditory/language processing. He is currently a
Postdoctoral Scholar with the Mental Processes and Brain Activation Lab,
INSERM U821, Bron, France.
Dr. Dalal is a member of the IEEE Engineering in Medicine and Biology
Society. He received the Young Investigator Award from the International Conference on Biomagnetism in 2004, the Ruth L. Kirschstein National ResearchService Award from the National Institute on Deafness and Other Communication Disorders, and the Chateaubriand Fellowship from the Embassy of France
in the U.S.

Srikantan S. Nagarajan (S’90–M’91–SM’07)
received the M.S. and Ph.D. degrees in biomedical
engineering from Case Western Reserve University(CWRU), Cleveland, OH.
His research career began at the Applied NeuralControl Laboratory, CWRU. After graduate school,
he did a postdoctoral fellowship at the Keck Center
for Integrative Neuroscience, University of California, San Francisco (UCSF), under the mentorshipof Drs. Michael Merzenich and Christoph Schreiner.
Subsequently, he was a tenure-track faculty memberin the Department of Bioengineering, University of Utah. Currently, he is the
Director of the Biomagnetic Imaging Laboratory, an Associate Professor in
Residence in the Department of Radiology, and a member in the UCSF/University of California, Berkeley Joint Graduate Program in Bioengineering.
His research interests are in the area of neural engineering where his goal is
to better understand dynamics of brain networks involved in processing, andlearning, of complex human behaviors such as speech, through the developmentof functional brain imaging technologies.