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Modified Beamformers for Coherent Source Region
Suppression
Sarang S. Dalal* , Student Member, IEEE, Kensuke Sekihara, Senior Member, IEEE, andSrikantan S. Nagarajan, Member, IEEE
Abstract—Many tomographic source localization algorithms
used in biomagnetic imaging assume, explicitly or sometimes
implicitly, that the source activity at different brain locations
are either independent or that the correlation structure between
sources is known. Among these algorithms is a class of adaptive
spatial filters known as beamformers, which have superior spatiotemporal resolution abilities. The performance of beamformers
is robust to weakly coherent sources. However, these algorithms
are extremely sensitive to the presence of strongly coherent
sources. A frequent mode of failure in beamformers occurs withreconstruction of auditory evoked fields (AEFs), in which bilateralauditory cortices are highly coherent in their activation. Here,
we present a novel beamformer that suppresses activation fromregions with interfering coherent sources. First, a volume containing the interfering sources is defined. The lead field matrix
for this volume is computed and reduced into a few significant
columns using singular value decomposition (SVD). A vector
beamformer is then constructed by rejecting the contribution ofsources in the suppression region while allowing for source reconstruction at other specified regions. Performance of this algorithm
was first validated with simulated data. Subsequent tests of thismodified beamformer were performed on bilateral AEF data.
An unmodified vector beamformer using whole head coverage
misplaces the source medially. After defining a suppression region
containing the temporal cortex on one side, the described method
consistently results in clear focal activations at expected regions of
the contralateral superior temporal plane.
Index Terms—Beamformer, biomagnetism, functional neuroimaging, magnetoencephalography, MEG inverse problems,
neuromagnetic signal processing.

I. INTRODUCTIONM

AGNETOENCEPHALOGRAPHY (MEG) is a functional neuroimaging technique with the ability to resolve
brain dynamics on the order of milliseconds [1]. In contrast
to other techniques, MEG provides a direct measure of thebrain’s neuronal activity that is relatively undistorted by thevarious tissue layers of the head [2]. The potential for improved
spatiotemporal reconstruction of neural activity has emerged

Manuscript received June 8, 2005; revised January 16, 2006. The work of S.
S. Dalal was supported in part by the National Institutes of Health (NIH) underGrant F31 DC006762. The work of S. S. Nagarajan was supported in part by
the NIH under Grant R01 DC004855-01A1. Asterisk indicates corresponding
author.
*S. S. Dalal is with the UCSF/Berkeley Joint Graduate Group in Bioengineering and the Department of Radiology, University of California, San Francisco, CA 94143-0628 USA (e-mail: sarang@hurricane.ucsf.edu).
K. Sekihara is with the Department of Systems Design and Engineering,Tokyo Metropolitan University, Tokyo 191-0065, Japan (e-mail: ksekiha@cc.
tmit.ac.jp).
S. S. Nagarajan is with the UCSF/Berkeley Joint Graduate Group in Bioengineering and the Department of Radiology, University of California, San Francisco, CA 94143-0628 USA (e-mail: sri@radiology.ucsf.edu).
Digital Object Identifier 10.1109/TBME.2006.873752

with the advent of dense MEG sensor arrays with whole-headcoverage.
Several source reconstruction algorithms, each employing
a different set of assumptions, have been proposed to overcome the ill-posed inverse problem. Source reconstructions
from MEG data can be classified as either parametric or tomographic. Parametric methods include equivalent current
dipole (ECD) fitting techniques; they often require knowledge
about the number of sources and their approximate locations,and poorly model sources with a large spatial extent. Tomographic methods reconstruct source activity at each voxel in thebrain. It has been shown that a class of adaptive spatial filters
known as beamformers [3] have the best spatial resolution
and performance amongst existing tomographic methods [4].
Beamformers avoid the high number of parameters and nonlinear nature of ECD analysis. Beamformers can also be used
to perform time-frequency analysis and study oscillatory powerchanges of neural sources, which are difficult to reconstructusing parametric methods [5]–[7].
However, the performance of beamformers degrades in
the presence of highly correlated sources. For instance, theeigenspace beamformer is robust to moderate source correlations; however, sources that are strongly correlated
are poorly resolved [8]. Whole-head coverage increases thelikelihood of encountering such a situation since bilateral activations are often strongly coherent. For example, beamformer
reconstructions of auditory evoked fields (AEFs) commonly
exhibit this failure, attenuating the two true sources from eachprimary auditory cortex and often erroneously placing a single
low-amplitude source centered between them.
A partial solution to this problem is to process left temporal
sensors separately from right temporal sensors; this method was
first employed to overcome difficulties encountered with ECD
methods [9] and later applied to beamformers [10]. However,this method is not entirely satisfactory since it essentially discards most of the information gained from a whole-head sensor
array; it also may not be sufficient for sources in closer proximity such as bilateral occipital activations evoked by visual
stimuli or other more complex configurations.
We propose a modified beamformer method that circumvents these shortcomings by defining a region to ignore andaugmenting the lead field matrix with this information. We
demonstrate the efficacy of the proposed modified beamformer
with simulations and real AEF data. Such a method improves
upon the accuracy and resolution of beamforming.
Throughout this paper, plain italics indicate scalars, lowercase boldface italics indicate vectors, and uppercase boldface
italics indicate matrices.

0018-9294/$20.00 © 2006 IEEE

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II. METHOD

The solution is known to be [13]A. Definitions and Problem Formulation
We shall define the magnetic field measured by theth detector coil at time as
and a column vector
as a set of measured data,is the total number of detector coils, and the superwhereindicates the matrix transpose. The spatial location
scriptsuch thatis represented by a three-dimensional vector.
To express the moment magnitudes of the sources located at
at time in the three orthogonal directions, we define.
To express the orientation of the th source, we define theangles between its moment vector and the , , and axes as, , andrespectively. The orientation of the th source is. We assume in this paper
defined as a vector
that the orientation of each source is time independent.
The second-order moment matrix of the measurement is de, i.e.,, whereindicates the ennoted
semble average. In practice, the ensemble average is often replaced with the time average over a certain time window. When,is also equal to the covariance matrix of the measurement.
The lead field vector for the component of a source atis defined as. Here,expresses the th sensor output induced by the unit-magnitude
source that is located at and oriented in the direction, wheresuch that. We define the lead field matrix as, representing the sensitivity of thesensor array at in all three orthogonal directions. Modeling thehead as a spherical volume conductor reduces the lead field totwo orthogonal components in spherical coordinates, and[11]. In this case, the orientation of the th source can instead, resulting in
be represented by a vector.
B. Conventional Adaptive Spatial Filtering
An adaptive spatial filter estimate of the source moment mais given by
trix

(4)
An eigenspace-projected minimum variance beamformer isobtained by projecting the weight matrix of the LCMV beamformer onto the signal subspace of the measurement covariancematrix(5)
wherecontains the eigenvectors representing the signal. This eigenspace projection overcomes thesubspace of
SNR degradation caused by array mismatch and smoothes timecourses of source reconstruction [12].
C. Proposed Adaptive Beamformer With Coherent Suppression
1) A Method for Point Source Interference Suppression: The
LCMV formulation allows one to add additional null constraints
to suppress the influence of correlated sources. If the exact location of an interfering source is known to be at point , then
the weight matrix can be reformulated(6)
Similarly, for the orthogonal orientation(7)
Therefore, the standard composite lead field matrix may be augmented with additional columns
(8)

(1)
and
, whereandare theweight vectors for the and directions, respectively, whileandare the corresponding components of thesource moment vector.
This section reviews an eigenspace-projected linearly constrained minimum variance vector (LCMV) beamformer [12],and[13]. Omitting from the expression, the weight vectors
are calculated with the following constraints:

resulting in a new weight matrix formulation

In (1),

(2)

(3)

(9)
2) A Method for Region Suppression: However, the precise
location of interference is seldom known a priori. Nevertheless,in many cases, an interfering source can safely be presumed tooriginate from somewhere within a larger brain region. For example, in auditory MEG experiments, interfering sources are
typically present in superior temporal areas contralateral to theregion of interest. It is, therefore, desirable to extend this formulation to suppress a selected region suspected to contain aninterfering source.
In order to accomplish this, the lead field of the desired suppression region may be added to the null constraint of the weightmatrix computation. As with a single point, the lead field ma-

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trix is augmented with the contribution from each voxel in thedesired suppression region,

(10)
where

is composed of the points

and. Again, fororientation(11), the weight matrix

Then, defining
retains a familiar form

Fig. 1. The x, y , and z coordinates used to express the reconstruction resultsin Section IV. The midpoint between the left and right preauricular points was
defined as the coordinate origin. The axis directed away from the origin to theleft preauricular point was defined as the y axis, and that from the origin to
the nasion was the x axis. The z axis was defined as the axis perpendicular
to both these axes and was directed from the origin toward the vertex.

+

+

+

(12)
In practice, will consist of several thousand voxels, making
a highly singular matrix and,the producttherefore, difficult to invert accurately. Furthermore, each additional column increases the computational load proportionately.
To remedy these two problems, singular value decomposition; the most significant components(SVD) may be applied to
can then be chosen to limit the number of additional columns
and improve the condition number of this matrix. This has theadded benefit of allowing the suppression region to be sampled
at a different spatial resolution from the region of interest;a coarser suppression grid would be more computationallyefficient without significantly compromising accuracy.
such that
, the
left
Let us define
singular vectors of, and. Then,building on the full eigenspace-projection beamformer, we conby performing Gram-Schmidt
struct an orthonormal basis. Finally,
orthogonalization on the composite matrix
forfrom (5), andforfromsubstituting(12), we obtain
(13)
as
and
We use the first two columns of-dimensional column vectors
andThat is, defining
such thatthe weight vectors,

and.

, are obtained using(14)

The normalized lead field matrix
may beused to prevent erroneous values near the local sphere origin[13], [14].
III. NUMERICAL EXPERIMENTS
A. Data Generation
A series of numerical experiments were conducted to test theeffectiveness of the proposed method. The sensor configurationof the 275-channel CTF Omega 2000 biomagnetic measurement
system (VSM MedTech, Coquitlam, BC, Canada) was used.

= 40 mmFig. 2. Above left, the reconstruction profile on the z
plane, using
the conventional beamformer. The circle and square indicate the locations of
the two simulated coherent sources. At right, the time series associated with thespatial peak at (8.0, 6.0, 40.0) mm.

Data were simulated and processed using a development version of NUTMEG [15].
Two identical synchronous sine wave sources were synthesized and placed at (0, 30, 40) mm and (0, 30, 40) mm, withcoordinates defined as in Fig. 1. A sensor lead field was calculated using a single-layer spherical volume conductor as the forward model [11] and the Omega 2000’s sensor geometry, with
2-mm grid spacing. Gaussian white noise was added to the generated data such that the signal-to-noise (SNR) was equal to 2.
The SNR was defined as the ratio of the Frobenius norm of the
MEG data matrix to that of the noise matrix.
The conventional beamformer erroneously placed a diffuse“source” centered at (8.0, 6.0, 40.0) mm, shown in Fig. 2. Notealso the corresponding time course does not show discernible
distortion or other indications of an inaccurate reconstruction,and can be misleading in applications to real data. Applying
point suppression to the coordinates of one source resulted ina highly focal and accurate reconstruction of the other source(Fig. 3).
Next, we synthesized a total of six sine wave sources to test
the region-based suppression technique. Four of these sourceswere synchronous, while the other two had different frequencies and phases. As expected, the conventional beamformer reconstructs the two unique sources, but fails to resolve any of
the four correlated sources (Fig. 4). A suppression region was
defined covering three of the four correlated sources, leaving
three sources in the region of interest; the region was

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Fig. 3. Above left, the reconstruction profile on the z = 40 mm plane, using
the beamformer with the point suppression modification and the same simulated
data as in Fig. 2. The left source was suppressed, perfectly reconstructing theright source. At right, the time series reconstructed for the right source.

Fig. 4. The reconstruction profile and time courses obtained from six simulated
sources, four of which are coherent (indicated by circles) and two of which
are uncorrelated (indicated by the square and the diamond). None of the four
coherent sources are resolved.

. The eigenspectrum of this region’s lead field is shown in
Fig. 5(a). Choosing eigenvectors that represent 95% of the variance is generally acceptable; however, often the precise eigenvalue thresholding is apparent by inspection. In this case, thefirst 13 eigenvectors, representing 93% of the variance, werechosen to augment the lead field of the reconstruction region
as a null constraint. As shown in Fig. 5(b), all three sources ofinterest were resolved; the peak of the reconstructed correlated
source had a localization error of 4.9 mm (approximately twovoxels), while the other two were reconstructed perfectly withno localization error.
isThe size of the pseudosignal subspace represented bygreater than the true signal subspace , hence a small increasein noise was observed in the reconstructed time series. To assess how noise is affected by the chosen eigenvectors, the meansquare errors (MSE) of the time courses were calculated forseveral choices (see Fig. 6). As expected, the MSE of the timecourse for the correlated source reduced with more eigenvalues,
leveling off at approximately 13. The error of one uncorrelated
source increased slightly with more eigenvalues, while the otherremained fairly constant.
To assess how well an interfering source is suppressed
depending on its position within the suppression region, we
returned to the case of two synchronous sine wave sources.

Fig. 5. (a) Eigenspectrum of the lead field of the suppression region. 13 eigenvalues, accounting for 93% of the total variance, were chosen by inspection. (b)The reconstruction profile and time courses obtained when defining the boxedarea as the suppression region. The four circles indicate the coherent sources,
while the sources indicated by the square and diamond are uncorrelated.

Fig. 6. Mean squared error of the time courses of the three sources as a functionof eigenvector components selected for the lead field of the suppression region.
The circles, diamonds, and squares correspond to the sources in Figs. 4 and 5.

Using the left hemisphere suppression region defined above,
one source was fixed at (0, 30,40) mm, while the algorithm

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Fig. 8. A bilateral AEF response to a 1-kHz pure tone, with data from sensors
over the left and right temporal cortices overlaid.

Fig. 7. Contour map showing reconstruction error (mm) of right source as afunction of position of left source within suppression region.

was evaluated with the other source at each of 190 locations
throughout the suppression region. Fig. 7 shows the localization
error of the source of interest as a function of the interfering
source within the suppression region. The mean error was
4.8 mm, with 73% of locations within the suppression region
yielding reconstruction errors of 5 mm or less. Interfering
sources placed within 5 mm of the edges farthest from themodel sphere center produced the largest errors of up to 20 mm.
IV. APPLICATION TO AUDITORY-EVOKED MEG DATA
In order to apply our proposed technique to real data, auditory
evoked field data was acquired from a 24-year-old female using
pure tones. Data was acquired with a 275-channel whole-head
MEG device from CTF Systems (VSM MedTech, Coquitlam,
BC, Canada). The recordings were collected in accordance withthe ethical standards of the UCSF Institutional Review Board
and Helsinki Declaration of 1975, as revised in 1983. The auditory stimuli consisted of 1-kHz pure tones of 400 ms duration. The interstimulus interval was randomly varied between
1.5–1.6 s. The sampling frequency was set at 1200 Hz. All postprocessing and analysis were performed using a development
version of NUTMEG [15]. A digital filter was used to highpass
the data at 1 Hz. After visual rejection of trials containing eyeblink and movement artifacts, a total of 112 trials were averaged(Fig. 8).

Fig. 9. (a) Conventional beamformer reconstruction of AEF data, exhibiting
failure due to correlated sources. The algorithm reconstructs a false source,
placed in the frontal lobe, superior and anterior to the expected location on
the superiotemporal plane. The spatial activation shown is at FWHM. The timecourse shown is for the spatial peak marked by the crosshairs. (b) Reconstruction of AEF data with suppression of the outlined region (left temporal cortex).
Right auditory cortex is revealed, along with its time course. (c) Similarly, region suppression of right temporal cortex reveals left auditory cortex.

Source reconstruction using a standard eigenspace vector
beamformer with the full 275-channel array shows a failure
typical of simultaneous bilateral activation, placing a low-amplitude, diffuse source bleeding towards the center of the model
sphere [Fig. 9(a)]. For this reconstruction, the signal subspacedimension was set to two because the eigenspectrum showed
two dominant eigenvalues. Note that the reconstructed time

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series appears to be reasonable and does not show obvious
symptoms of failure.
Applying the proposed technique, we used all 275 channels
and defined the suppression region to be a broad volumecontaining right temporal areas. We then projected
the lead field of the suppression region onto the 13 dominanteigenvectors and completed source reconstruction. As shown in
Fig. 9(b), a plausible location for left primary auditory cortex on
the superior temporal plane clearly emerges. Similarly, selecting
a similar suppression region containing left temporal areas results in a peak at a plausible location for right primary auditory
cortex [Fig. 9(c)]. Furthermore, no spurious activations near thecenter of the model sphere were observed. The correlation coefficient between the reconstructed time series for the two peaks
was observed to be 0.90.
For comparison, the data were also fit to a spatiotemporal
ECD model with DipoleFit (VSM MedTech, Coquitlam, BC,
Canada). Using the same single-sphere head model as the beamformer analyses, two dipoles were fit simultaneously over theinterval 75 ms to 100 ms, achieving a total weighted error of
9.8%. The left dipole localized 12.5 mm lateral and inferior to
the left source reconstructed by the modified beamformer, while
the right dipole localized 6.4 mm superior and slightly medial
to the reconstructed right source. Both of these locations were
within the full-width at half-maximum (FWHM) of each peak of
the modified beamformer. The correlation coefficient between
the two dipole moments was 0.93.

V. CONCLUSION
Adaptive beamformers have been shown to have zero bias
and the highest spatial resolution amongst various spatial
filtering methods used in neuromagnetic source reconstruction[16]. However, the accurate resolution of highly correlated
sources has been problematic for beamformer techniques. One
workaround has been to simply process different sensor groups
independently, especially for auditory experiments. While such
results may be satisfactory for some situations, this method
likely will not be sufficient for correlated sources that are in
the same hemisphere or otherwise closer together. Hence,highly correlated visual or somatosensory activity may not be
accurately resolved by simply ignoring distant sensors.
The described method for suppressing regions of coherent
activation is an important development for MEG inverse
techniques. We have presented a modification that solves the
problem of coherent sources when the approximate region of
one source is known and disjoint from the other. This method
essentially allows for a specific exception to the assumption of
independent sources commonly made in beamformer reconstructions. Thus, simultaneous bilateral activations may now
be accurately reconstructed using this technique without discarding any channels. It may also provide a solution in the caseof highly correlated sources located in the same hemisphere orotherwise close to each other.
While the manual selection of the suppression region is adrawback of the method proposed here, the suppression region
can be defined with the aid of a priori knowledge, potentially

including information from functional magnetic resonanceimaging studies. It may also be possible to automate the selection through a method that scans through several potential
suppression regions.
ACKNOWLEDGMENT
The authors would like to thank A. M. Findlay and S. M.
Honma for their assistance in acquiring MEG data, as well as
J. M. Zumer and K. E. Hild for helpful discussions.
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DALAL et al.: MODIFIED BEAMFORMERS FOR COHERENT SOURCE REGION SUPPRESSION

Sarang S. Dalal (S’99) was born in Hayward, CA,
in 1978. He received the B.S. degree in biomedical
engineering, with a concentration in electrical engineering and a minor in psychology, from the Johns
Hopkins University, Baltimore, MD, in 2000. He iscurrently working towards the Ph.D. degree in bioengineering, based jointly at the University of California, San Francisco (UCSF) and the University ofCalifornia, Berkeley.
His research interests include neuromagnetic
source reconstruction, time-frequency analyses,
cortical connectivity, and auditory/language processing.
Mr. 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 (2004) and the Ruth L. Kirschstein National ResearchService Award from the National Institute on Deafness and Other Communication Disorders (NIDCD).

Kensuke Sekihara (M’88–SM’06) received the
M.S. and Ph.D. degrees from Tokyo Institute ofTechnology, Tokyo, Japan, in 1976 and 1987,
respectively.
From 1976 to 2000, he worked with the Central
Research Laboratory, Hitachi, Ltd., Tokyo. He wasa Visiting Research Scientist at Stanford University,
California, Stanford, from 1985 to 1986, and at
Basic Development, Siemens Medical Engineering,
Erlangen, Germany, from 1991 to 1992. From 1996
to 2000, he worked with the “Mind Articulation”

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research project sponsored by the Japan Science and Technology Corporation. He is currently a Professor at the Department of Systems Design &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 the IEEE Engineering in Medicine and Biology
Society and the IEEE Signal Processing Society.

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