www.elsevier.com/locate/ynimg
NeuroImage 25 (2005) 1056 – 1067

Localization bias and spatial resolution of adaptive and non-adaptive
spatial filters for MEG source reconstruction
Kensuke Sekihara,a,* Maneesh Sahani,b and Srikantan S. Nagarajan caDepartment of Electronic Systems and Engineering, Tokyo Metropolitan Institute of Technology, Asahigaoka 6-6, Hino, Tokyo 191-0065, Japan
Gatsby Computational Neuroscience Unit, University College London, Alexandra House, 17 Queen Square, London WC1N 3AR, UKc
Department of Radiology, University of California, San Francisco, 513 Parnassus Avenue, S362, San Francisco, CA 94143, USAb

Received 24 June 2004; revised 19 October 2004; accepted 30 November 2004
Available online 13 March 2005

This paper discusses the location bias and the spatial resolution in thereconstruction of a single dipole source by various spatial filtering
techniques used for neuromagnetic imaging. We first analyze thelocation bias for several representative adaptive and non-adaptive
spatial filters using their resolution kernels. This analysis theoreticallyvalidates previously reported empirical findings that standardized lowresolution electromagnetic tomography (sLORETA) has no location
bias. We also find that the minimum-variance spatial filter does exhibit
bias in the reconstructed location of a single source, but that this bias iseliminated by using the normalized lead field. We then focus on thecomparison of sLORETA and the lead-field normalized minimumvariance spatial filter, and analyze the effect of noise on source location
bias. We find that the signal-to-noise ratio (SNR) in the measurements
determines whether the sLORETA reconstruction has source locationbias, while the lead-field normalized minimum-variance spatial filter
has no location bias even in the presence of noise. Finally, we compare
the spatial resolution for sLORETA and the minimum-variance filter,and show that the minimum-variance filter attains much higher
resolution than sLORETA does. The results of these analyses arevalidated by numerical experiments as well as by reconstructions basedon two sets of evoked magnetic responses.
D 2004 Elsevier Inc. All rights reserved.
Keywords: Magnetoencephalography; Source reconstruction; Adaptive
spatial filter; Non-adaptive spatial filter; Resolution kernel

Introduction
Among the various technologies for noninvasive neural
measurement, the major advantage of magnetoencephalography(MEG) is its ability to provide fine temporal resolution, in the orderof milliseconds (Hämäläinen et al., 1993). In neuromagnetic* Corresponding author. Fax: +81 42 585 8642.
E-mail address: ksekiha@cc.tmit.ac.jp (K. Sekihara).
Available online on ScienceDirect (www.sciencedirect.com).

imaging, the signals recorded by MEG sensors are resolved intofunctional maps of dynamic brain activity, while retaining this hightemporal precision. Many studies have addressed the question of
how this reconstruction can be done efficiently (Baillet et al.,2001).
One prominent class of techniques, known as adaptive spatial
filtering, was originally developed in the field of array signal
processing (van Veen and Buckley, 1988). A well-known exampleof this class, the minimum-variance spatial filter, has beensuccessfully applied to MEG/EEG source reconstruction problems(Robinson and Vrba, 1999; Sekihara and Scholz, 1996; Sekihara etal., 2001, 2002a; van Veen et al., 1997). Another popular class ofsource reconstruction methods is that of minimum-norm-based
tomographic reconstruction. This class includes the original
minimum-norm method (Hämäläinen and Ilmoniemi, 1984), theweight-normalized minimum-norm method (Dale et al., 2000), anda recently proposed method called standardized low-resolution
electromagnetic tomography (sLORETA) (Pascual-Marqui, 2002).
This paper formulates these minimum-norm-based methods as
non-adaptive spatial filters, and compares them with the adaptive
minimum-variance spatial filter on a unified basis.
We first compare several representative (either adaptive or nonadaptive) spatial filters with respect to the bias they introduce in thereconstructed location of a single source in the absence of noise. It
is shown that under these conditions sLORETA has no location
bias. This fact has been found previously in computer simulations(Pascual-Marqui, 2002), and our analysis validates these empiricalfindings. We also show that the minimum-variance spatial filter
does lead to biased source reconstructions, but that this bias can beeliminated by using the normalized lead field. We then focus on thebias-free two methods, sLORETA and the lead-field normalized
minimum-variance filter, and analyze how they perform with noisy
measurements. We find that, depending on the signal-to-noise ratio(SNR), the sLORETA reconstruction may have some sourcelocation bias. The lead-field normalized minimum-variance filter,
however, has no location bias even in the presence of noise.
Finally, we compare the spatial resolution of the two methods and1053-8119/$ - see front matter D 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2004.11.051

YNIMG-02926; No. of pages: 12; 4C: 7, 9

K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

show that the minimum-variance filter generally attains muchhigher spatial resolution than sLORETA does.
Following a brief introduction of several representative nonadaptive and adaptive spatial filter formulations, we analyze theirsource-location biases in Bias for the estimated source location andEffects of noise on the location bias. Spatial resolution presents thespatial resolution comparison for sLORETA and the minimumvariance filter. Numerical experiments presents a series ofnumerical experiments that validate the arguments in Theory. In
Comparison using evoked MEG data, we compare sLORETA andthe minimum-variance filter by applying them to somatosensoryand auditory-evoked MEG data.

the minimum-norm spatial filter (Hämäläinen and Ilmoniemi,
1984). Its weight vector is given bywðrÞ ¼ G 1 l ðrÞ:ð3ÞThe matrix G, often referred to as the gram matrix, is defined such
that
Z


G¼
LðrÞLT ðrÞ dr:
ð4Þ
One variant of the minimum-norm filter is the weightnormalized minimum-norm filter (Dale et al., 2000) whose weights
are given byG1 l ðrÞwðrÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :l T ðrÞG2 l ðrÞ

Theory1057

ð5Þ

Spatial filter formulation
Definitions
We define the magnetic field measured by the mth detector
coil at time t as b m (t), and a column vector b(t) = [b 1 (t), b 2(t),. . ., b M (t)]T as a set of measured data where M is the total
number of sensor coils and superscript T indicates the matrix
transpose. The covariance matrix of the measurement is denoted
Rb , that is, Rb = hb(t)b T (t)i where hdi indicates the ensemble
average. (This matrix is actually equal to the second-ordermoment matrix but is customarily called the covariance matrix.)
The spatial location is represented by a three-dimensional vector
r:r = (x, y, z). The source moment magnitude at r and time t isdenoted as s(r, t). The orientation of a source at r is defined as athree-dimensional vector h(r) = [g x (r), g y (r), g z (r)]T whose fcomponent (where f equals x, y, or z) is equal to the cosine of theangle between the direction of the source moment and the fdirection.
We define l mf (r) as the output of the mth sensor; the output is
induced by the unit-magnitude source located at r and directed in
the f direction. The column vector l f (r) is defined as l f (r) = [l 1f(r),l 2f (r),. . ., l Mf (r)]T. The lead field matrix, which represents thesensitivity of the whole sensor array at r, is defined as L(r) = [l x (r),l y (r),l z (r)]. The lead-field vector in the source-moment direction isdefined as l(r) where l(r) = L(r) h(r). Using the superposition law,
the relationship between b(t) and s(r, t) is expressed asZZbðt Þ ¼LðrÞhðrÞsðr; t Þdr þ nðtÞ ¼l ðrÞsðr; t Þdr þ nðt Þ: ð1Þ
Here, n (t) is the additive noise. The spatial filter uses a simple
linear operation for estimating the magnitude of source activities,
such thatŝs ðr; t Þ ¼ wT ðrÞbðt Þ;

ð2Þ

where ŝ(r, t) is the estimate of the source magnitude, and a column
vector w (r) is a set of the spatial-filter weights, which characterizes each spatial filter.
Adaptive and non-adaptive spatial filters
There are two types of spatial filters. One is a non-adaptive
spatial filter in which the filter weights are independent from themeasurements. The L 2-norm based tomographic reconstruction
methods can in principle be interpreted as non-adaptive spatial
filters. The most basic and well-known non-adaptive spatial filter is

Another variant of the minimum-norm filter, called sLORETA, hasbeen recently proposed (Pascual-Marqui, 2002). The weight vector
for sLORETA isG1 l ðrÞwðrÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :l T ðrÞG1 l ðrÞ

ð6Þ

The derivations of the above-mentioned filter weights arepresented in Appendix B. It should be pointed out that because
the gram matrix G is close to a singular matrix for a typicalneuromagnetic measurement geometry, the regularization may be
applied when calculating G1. That is, (G + cI)1 is calculated
instead of calculating G1. The choice of regularization parameterc has been the subject of many investigations (Hansen, 1998). The
value of the regularization parameter is typically set to the averagevariance of the sensor noise.
An adaptive spatial filter uses weights that depend on themeasurements. The best-known adaptive spatial filter is theminimum-variance spatial filter (Robinson and Vrba, 1999; van
Veen et al., 1997); its weight vector is expressed asR 1 l ðrÞ:wðrÞ ¼  T b 1l ðrÞRb l ðrÞ

ð7Þ

In Eqs. (3)–(7), the lead field vector l(r), which is the lead field inthe source-moment direction h(r), is used for deriving the filter
weights. Here, since h(r) is generally unknown, the optimum
direction h opt(r), which gives the maximum filter output, can beused instead of the unknown h(r). That is, l(r) can be obtainedusing l(r) = L(r)h opt(r) (Sekihara and Scholz, 1996).
Resolution kernel
To compare various spatial filter formulations, we need a tool tocharacterize how appropriately the weights have been chosen. The
resolution kernel (Backus and Gilbert, 1968; de Peralta Menendez
et al., 1997) can play this role. We introduce the resolution kernel,
derived by combining Eqs. (1) and (2),ZZŝs ðrÞ ¼
wT ðrÞl ðr VÞsðr Þdr
V V¼
Rðr; r VÞsðr VÞdr ;V
ð8Þ
where
Rðr; r VÞ ¼ wT ðrÞl ðr VÞ:

ð9Þ

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K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

Here, the explicit notation of time t is omitted. This Rðr; r VÞ iscalled the resolution kernel. It expresses the relationship between
the original and reconstructed source distributions. When a single
source with a unit magnitude exists at r 1, we can substitute s(r) =d (r  r 1) into Eq. (8), and write the reconstructed results asŝs ðrÞ ¼ Rðr; r1 Þ. Thus, the resolution kernel directly gives thespatial reconstruction that results from a single point source in theabsence of noise.
Defining f as f = l(r 1), and substituting Eq. (3) into Eq. (9), theresolution kernel for the minimum-norm spatial filter is given by˜
Rwhere f =Tf / t f t and G̃2is the gram matrix obtained using G̃ =[L (r) L (r)] / t L (r)t dr. There is no guarantee that the lefthand side of the second equation in Eq. (16) always has a positive
value, because the signs may be different between a component ofa vector f˜  l̃ and the corresponding component of f. Therefore, it
is clear that the minimum-norm method generally has a localization
bias even with the normalized lead field—although it has no bias insome locations where Eq. (16) is coincidentally satisfied.
For the weight-normalized minimum-norm filter, the condition
in Eq. (14) is expressed as

Rðr; r1 Þ ¼ l T G1 f ;f T G1 f
l T G1 f
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :
T 2
f G f
l T G2 l

ð10Þ

where the explicit notation of r is omitted from l(r). Using Eq. (5),the resolution kernel of the weight-normalized minimum-norm
spatial filter isl T G1 fﬃ:
Rðr; r1 Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
l T G2 l
The resolution kernel for sLORETA is given by
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ l T G1 f
Rðr; r1 Þ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ f T G1 f cos l; f j G1 ;
T 1
l G l

ð11Þ

ð12Þ

where the generalized cosine is defined according to Eq. (41).
Using Eq. (7), the resolution kernel for minimum-variance filter isexpressed as
Rðr; r1 Þ ¼

l T R1b f:
l T R1b l

ð13Þ

An important property for the resolution kernel is to have its
maximum at the source location. If this is not the case, thereconstructed source distribution will contain systematic biases andmay thus be totally different from the true source distribution. In
the following subsection, we check the resolution kernels for eachof the spatial filters described in Spatial filter formulation, to see
whether the corresponding filter avoids spatial bias.
Bias for the estimated source location
The bias for the estimated source location is evaluated bychecking whether the resolution kernel has its maximum at thesource location r 1, that is, whether the condition below holds forany r (r p r 1) in the reconstruction region
Rðr1 ; r1 Þ N Rðr; r1 Þ:

ð14Þ

For the minimum-norm filter, this condition is expressed asf T G1 f N l T G1 f :ð15Þ

Because the norm of l becomes very large in a region close to thesensors, the above inequality obviously does not hold. Actually, it
is well known that in the minimum-norm filter, the sourcereconstruction is severely biased toward the sensor array (Jeffs etal., 1987). Because of this fact, when applying the minimum-norm
filter, the normalized lead field vector l̃ = l / tl t is often used. The
condition for no location bias in this case isT 1
f˜ T G̃G 1 f N l̃l T G̃
G 1 f or ff̃  l̃l G̃
G f N0ð16Þð17Þ

It is not straightforward to see whether this inequality holds
for any pair of l and f. However, we can numerically showthat it does not generally hold, and we give, in Numericalexperiments, a typical numerical example of biased sourcereconstruction results obtained from this weight-normalized minimum-norm filter. For sLORETA, using Eq. (12), the condition in
Eq. (14) is expressed ascos l; f j G1 b 1:
ð18Þ
Since G1 is a positive definite matrix, this condition holds for any
f and l ( f p l), as mentioned in Appendix A. It is empiricallyknown that sLORETA has no location bias for a single source(Pascual-Marqui, 2002), and the analysis presented here validates
this empirical observation.
For the minimum-variance spatial filter, the condition forhaving no location bias isl T R1b f
b 1:
l T R1b l

ð19Þ

Using Eqs. (39) and (40) in Appendix A, the above condition canbe rewritten asl T R1
tf tcosðl; f Þ
b f
b 1:
¼
tlt
½
1
þ
a
1  cos2 ðl; f Þll T R1b

ð20Þ

In the above equation, a = (r 12 / r 02)t f t2, where r 12 and r 02 are thepower of source activity and noise power, as defined in Appendix
A. This a is often referred to as the input power SNR. It can beseen that the inequality in Eq. (20) may not hold when tlt issmall. This can happen in a region near the center of the spherewhen the spherical homogeneous conductor model (Sarvas, 1987)
is used to derive the lead field. As a result, severe artifacts appear
near the center of the sphere. However, the use of the normalized
lead field vector l̃ can avoid these artifacts, and the condition for nolocation bias in this case is expressed ascos l̃l ; fll̃ T R1b fRðr; r1 Þ ¼ T 1 ¼ t f tl̃l Rb l̃l
1 þ a 1  cos2 l̃l ; fb Rðr1 ; r1 Þ ¼ t f t:

ð21Þ

Since a has a positive value, it is clear that this inequality is always
fulfilled for any f and l, and we can conclude that the minimumvariance spatial filter with the normalized lead field has no location
bias.

K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

1059

such analysis should be valid when the SNR is high. However,when the SNR is low, the noise may cause bias in the estimatedsource location even though the resolution kernel has its maximum
at the true source location. The output signal power at a location r isequal to r21 Rðr; r1 Þ2 , and the output noise power is equal to r 02tw(r)t2. Therefore, the condition for no location bias is
r21 Rðr1 ; r1 Þ2 þ r20 twðr1 Þt2 N r21 Rðr; r1 Þ2 þ r20 twðrÞt2 :ð22Þ

This condition can be written as
Rðr; r1 Þ2 ½1 þ XðrÞ=ab 1;
Rðr1 ; r1 Þ2 ½1 þ Xðr1 Þ=að23Þwhere a is the input SNR and X(r) is given by
X ðr Þ ¼
Fig. 1. The coordinate system used in the numerical experiments. The
coordinate origin was set at the center of the sensor coil located at the centerof the array. The plane at x = 0 cm is shown. The circle indicates theboundary of the sphere used for the forward calculations. The center of thesphere was set to (0, 0, 12).

tf t 2 twðrÞt2
Rðr; r1 Þ2

:

ð24Þ

For sLORETA, using Eqs. (6), (12), and (24), Eq. (23) isrewritten as1 þ XðrÞ=acos2 l; f jG1 b 1:
1 þ Xðr1 Þ=að25Þ

Effects of noise on the location bias
In the preceding section, the location bias in the sourcereconstruction has been analyzed using the resolution kernel andIt can be seen in Eq. (25) that when a is very large, this condition
becomes identical to Eq. (18). However, in general, the value of adetermines whether the condition in Eq. (25) is satisfied for any lFig. 2. The plot of the square of the resolution kernel Rðr; r1 Þ2 on the plane x = 0. The point source is located at (0, 1.5, 6) cm. The contour plots in the lefthand side from top to bottom, respectively, show the results of the minimum-norm filter, the lead-field normalized minimum-norm filter, and the weightnormalized minimum-norm filter. The plots in the right-hand side from the top to the bottom, respectively, show the results of sLORETA, the minimumvariance filter, and the lead-field normalized minimum-variance filter. The cross mark indicates the source location. Each contour map is normalized by its
maximum value and contains the same number of contour lines.

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K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

Fig. 3. The results of the experiments regarding the effects of noise on location bias. (a) Results of point-source reconstruction from sLORETA and (b) thosefrom the lead-field normalized minimum-variance filter. The point source is located at (0, 1.5, 6) cm. The contour plots from top to bottom, respectively,
show the results for the input SNR a equal to 8M, 4M, and M. The cross markindicates the source location. Each contour map is normalized by its
maximum value and contains the same number of contour lines.

and f. In our numerical experiments, we present examples of cases
in which this condition is either fulfilled or not fulfilled.
For the lead-field normalized minimum-variance spatial filter,
using the weight in Eq. (7) and the resolution kernel in Eq. (21), thecondition for no location bias is
cos2 l̃l ; f½1 þ XðrÞ=a
ð26Þ



 b 1;½1 þ Xðr1 Þ=a 1 þ a 1  cos2 l̃l ; f 2

the smoothed version of the true source distribution, and the mainlobe width of the resolution kernel can be a measure of the spatial
resolution. To compare the main-lobe width of the kernel, thepoint-spread function, f(r), is defined as the resolution kernel
normalized by its peak value: fðrÞ ¼ Rðr; r1 Þ=Rðr1 ; r1 Þ. For
sLORETA, f(r) is expressed asð29ÞfðrÞ ¼ cos l; f j G1 :where X(r) in this case is given by
tf t2 l̃l T R2
b l̃l
X ðr Þ ¼ 
2 :l̃l T R1b f

For the lead-field normalized minimum-variance spatial filter, it
is expressed as
ð27Þ
f ðr Þ ¼

Using Eqs. (37) and (38), the condition in Eq. (26) can be
simplified to1
 b 1;
1 þ a 1  cos2 l̃l ; fð28ÞThis relationship holds for any l̃ and f because a has a positive
value, and this fact indicates that the lead-field normalized
minimum-variance spatial filter has no location bias even when
the SNR is low.


cos l̃l ; fcosðl; f Þ


 ¼
:
1 þ a½1  cos2 ðl; f Þ
1 þ a 1  cos2 l̃l ; f

ð30Þ

The above equation shows that the spread function strongly
depends on the input SNR a. This dependency has been reported
previously (Cox, 1973; Gross et al., 2003). The major difference
between the sLORETA point-spread function and that of theminimum-variance filter is the denominator in Eq. (30). Because ausually has a value greater than the number of sensors1 M, thedenominator causes a rapid decay of the point spread function.
Consequently, the spatial resolution of the minimum-variance filter

Spatial resolution
1

When the resolution kernel has its maximum at the source
location, the reconstructed source distribution can be interpreted as

Note that since the field power average across sensor channels is
equal to r 21|| f ||2/M, a source whose average field power is equal to theaverage noise power has a equal to M.

K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

Fig. 4. The plot of the point-spread function (the normalized resolution
kernel) in the horizontal ( y) direction. The three solid lines indicate thespread-function of the minimum-variance filter for the cases of a = 8M,
a = 4M, and a = M. The broken line indicates the point-spread functionof sLORETA. The point source was assumed to exist at (0, 0, 6), andthe abscissa expresses the distance from the source location in the y
direction.

is usually much higher than that of sLORETA. In Numericalexperiments, numerical examples of the point-spread functions arepresented to demonstrate the high spatial resolution of theminimum-variance filter.
Numerical experiments
We used the coil configuration of the Magnes 2500k neuromagnetometer (4D Neuroimaging, San Diego) where 148 sensors
are arranged on a helmet-shaped surface. As shown in Fig. 1, thecoordinate origin was chosen as the center of the sensor array. The
z direction is defined as the direction perpendicular to the plane of
the coil located at this center. The x direction is defined as thatfrom the posterior to the anterior, and the y direction is defined asthat from the left to the right hemisphere. The values of (x, y, z) are
expressed in centimeters.
We first compare the location biases for the spatial filters
described in Spatial filter formulation. To plot the resolution
kernel, we assume that a point source exists at r 1 = (0, 1.5, 6).
The power of the resolution kernel R2 ðr; r1 Þ was plotted on the1061

plane, x = 0. The results are shown in Fig. 2. Here, the contour
plots on the left show, from top to bottom, respectively, the resultsof the minimum-norm filter, the minimum-norm filter with thenormalized lead field, and the weight-normalized minimum-norm
filter. The plots on the right show, from top to bottom, respectively,
sLORETA, the minimum-variance filter, and the lead-fieldnormalized minimum-variance filter. When deriving the weightvectors for these non-adaptive spatial filters, the gram matrix G
was obtained by numerically integrating Eq. (4) over the volume
defined as 5 V x V 5, 5 V y V 5, and 3 V z V 11, and the
Tikhonov regularization was used for inverting G with theregularization constant set to 106 k max, where k max is the
maximum eigenvalue of G. When deriving the minimum-varianceresults, the covariance matrix Rb was obtained using Eq. (36) witha = M where M is the total number of sensors and is equal to 148
in this computer simulation. Throughout these numerical experiments, the noise is assumed to be white Gaussian, and uncorrelated
between sensor channels.
The resolution kernel for the minimum-norm filter, either withor without the lead-field normalization, shows a severe location
bias toward the sensors, as mentioned in Bias for the estimatedsource location. The resolution kernel for the weight-normalized
minimum-norm spatial filter has its maximum at a wrong location
although the bias is not as large as that of the original minimumnorm method. The results clearly indicate that the relationship in
Eq. (17) does not generally hold. The resolution kernel ofsLORETA has its maximum at the source location. The resolution
kernel for the minimum-variance spatial filter without the lead fieldnormalization has two peaks: one at the source location and theother at the sphere origin, which is located at (0, 0, 12). The peakintensity at the sphere origin is much higher than that at the source
location. The resolution kernel for the minimum-variance filter
with the lead field normalization has its maximum at the source
location. These results validate the arguments in Bias for theestimated source location.
We next tested the validity of the arguments regarding the effects
of noise on the location bias. Assuming that a point source exists at(0, 1.5, 6), the power output hŝ(r)2i is calculated using hŝ(r)2i =w T (r) Rb w (r). Here, the three input SNRs were tested: a = 8M, a =4M, and a = M where M = 148. (As mentioned in Spatial resolution,
the definition of a is given by a = (r 12 / r 02)t f t2. We may use a
different definition of the input SNR f that is closer to thedefinition
conventionally used in MEG measurements, such thatqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 2
f¼
r1 t f t2 =M =r20 . Namely, in this definition, f is equal to
the square root of the field power average across sensor channels

Fig. 5. The averaged somatosensory response obtained using a 160-channel whole head sensor array (MEGVISION, Yokogawa Electric Corp, Tokyo, Japan)
with an electric stimulus. The stimulus was delivered to the right median nerve at the subject’s wrist with a 250-ms interstimulus interval. An epoch was
digitized at 10 kHz sampling frequency and a total of 10,000 epochs were averaged. The recorded epochs were filtered with a bandpass filter of 3–300 Hz
bandwidth.

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K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

dividedpﬃﬃﬃ by the average noise power. Then, f is equal to 1, 2, and2 2 when a = M, a = 4M, and a = 8M, respectively).
The reconstructed results of this point source obtained fromsLORETA are shown in Fig. 3a. The results from the lead-fieldnormalized minimum-variance filter are shown in Fig. 3b. In Fig.
3, the contour maps from top to bottom, respectively, show theresults for a = 8M, a = 4M, and a = M. These results show thatsLORETA can detect the source at the correct location for the twocases of higher SNRs. It, however, cannot detect the source whena = M, that is, when the input SNR is the lowest among the threecases. The results here validate the conclusion that the SNR

determines whether sLORETA reconstruction has the bias for theestimated source location. The results of the minimum-variancespatial filter show that, for all three SNR cases, it can detect thesource at the correct location.
The spatial resolution is next compared between sLORETA andthe minimum-variance spatial filter. The point-spread functions f(r)are plotted with respect to |r 1  r| where r 1 = (0, 0, 6). The plotof the point-spread function with respect to the y (horizontal)
direction is shown in Fig. 4. In this figure, the three solid lines
indicate the spread-function of the minimum-variance filter forthe cases of a = 8M, a = 4M, and a = M. The broken line

Fig. 6. (a) The reconstructed source-magnitude map at latencies of 15.5, 16.5, and 18.5 ms obtained using sLORETA. (b) The reconstructed source-magnitude
map obtained using the lead-field normalized minimum-variance filter at the same latencies. The maximum intensity projections of the reconstructed threedimensional current density onto the axial, coronal, and sagittal planes are shown in the right, middle and left panels, respectively. The letters L and R show
the left and right hemisphere, respectively. The circles depicting a human head indicate the projections of the sphere used in the forward calculations. Eachcontour map is normalized by its maximum value, and contains the same number of contour lines. The color of the contour lines shows the relative intensity
indicated by the color bar.

K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

1063

Fig. 7. The average evoked magnetic responses recorded with simultaneously applied auditory and somatosensory stimuli using the 37-channel Magnesk
magnetometer. The auditory stimulus was a 1000 Hz pure tone with a 200-ms duration; it was applied to the subject’s right ear. The somatosensory
stimulus was a 30-ms duration tactile pulse (17 psi) delivered to the distal segment of the right index finger. The data was acquired at a sampling frequencyof 1 kHz and averaged for 256 epochs. An on-line bandpass filter with a bandwidth from 1 to 400 Hz was used and no post-processing digital filter wasapplied.

indicates the kernel shape of sLORETA. These plots indicate thatthe spatial resolution of the minimum-variance filter is muchhigher than that of sLORETA. In the next section, sLORETA andthe minimum-variance spatial filter are applied to two sets of
evoked recordings to further demonstrate the difference in thespatial resolution between these two kinds of spatial filters.
Comparison using evoked MEG data
The somatosensory response was recorded using a 160-channel
whole head sensor array (MEGVISION, Yokogawa Electric Corp,
Tokyo, Japan) with an electric stimulus delivered to the right median
nerve at the subject’s wrist with a 250-ms interstimulus interval. An
epoch of 150 ms duration (50 ms pre- and 150 ms post-stimulus) was
digitized at 10 kHz sampling frequency and a total of 10,000 epochs
were averaged. The averaged somatosensory response is shown in
Fig. 5 where the primary N20m peak is clearly identified.
We first applied sLORETA to this somatosensory response.
When calculating the filter weight in Eq. (6), the gram matrix was
calculated by averaging L(r) LT (r) over the whole brain region.
We selected three time points, 15.5, 16.5, and 18.5 ms, near thevertex of the N20m peak. The reconstructed source-magnitude map
at these time points obtained from sLORETA is shown in Fig. 6a.
In this figure, the maximum intensity projection of the threedimensional current density reconstruction ŝ (r, t)2 onto the axial,
coronal, and sagittal planes are shown in the right, middle, and left
panels, respectively. Also, in this figure, the results in the upper,
middle, and bottom rows, correspond respectively to the reconstruction at the latencies of 15, 16.5, and 18.5 ms. These results
clearly indicate the activation of the left parietal region, probably
near the primary somatosensory (SI) area. In these reconstruction
results, however, there are no significant differences among theresults at these three latencies.
We next applied the minimum-variance spatial filter to thissomatosensory response. In these experiments, the weight vectorof the lead-field normalized minimum-variance filter is obtained
and then the weight vector is projected onto the signal subspaceof the measurement covariance matrix Rb . The resultant projected
weight is used for the spatial filtering. This projection can avoid
the SNR degradation caused by errors in the forward modeling orin estimating Rb from data samples (Sekihara et al., 2002). The

sample covariance matrix was calculated using a time window
between 12 and 96 ms. The reconstructed results at the same threelatencies are shown in Fig. 6b. Compared to the results in Fig. 6a,the results in Fig. 6b exhibit much higher spatial resolution.
Indeed, in Fig. 6b, two sources in the SI area can be resolved, andwe can observe the lateral one becoming stronger while themedial source becomes weaker during the period between 15.5
and 18.5 ms2. The comparison between Figs. 6(a) and (b) clearly
demonstrates the high spatial resolution of the minimum-variancespatial filter.
The second data set consists of evoked magnetic responses
recorded with simultaneously applied auditory and somatosensory
stimuli using the 37-channel Magnesk magnetometer (4D
Neuroimaging, San Diego). In these measurements, we applied
auditory and somatosensory stimuli with simultaneous onsets. The
auditory stimulus was a 1000-Hz pure tone with a 200-ms
duration; it was applied to the subject’s right ear. The somatosensory stimulus was a 30 ms-duration tactile pulse (17 psi)delivered to the distal segment of the right index finger. The data
were acquired at a sampling frequency of 1 kHz for theprestimulus interval of 300 ms and the poststimulus interval of
800 ms and averaged for 256 epochs. The sensor array was placedon the left hemisphere and positioned to best record the M100
auditory response. The mean interstimulus interval was 2 s,
randomly varied between 1.75 and 2.25 s. The measured magnetic
responses are shown in Fig. 7.
The results of the source reconstruction from sLORETA and theminimum-variance spatial filter are, respectively, shown in Figs. 8aand b for the selected three time points of 60 ms, 120 ms, and 180
ms. In the minimum-variance filter reconstruction, the covariancematrix of the measurements, Rb , was calculated from the time
window between 0 and 300 ms. The results for sLORETA in Fig.
8a cannot resolve the two sources in the primary somatosensory
and the primary auditory cortices. However, the results from thelead-field normalized minimum-variance filter in Fig. 8b resolvethese two sources. These results again demonstrate that the spatial

2
The physiological interpretation for such behavior of the two sourcesin the primary somatosensory cortex has been reported in (Hashimoto et al.,2001a,b).

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K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

Fig. 8. (a) The reconstructed source-magnitude map at latencies of 60, 120, and 180 ms obtained using sLORETA. (b) The reconstructed source-magnitude map
obtained using the lead-field normalized minimum-variance filter at the same latencies. The maximum intensity projections of the three-dimensional current
density reconstruction onto the axial, coronal, and sagittal planes are shown in the right, middle and left panels, respectively. The letters L and R show the left
and right hemisphere, respectively. The circles depicting a human head indicate the projections of the sphere used in the forward calculations. Each contour
map is normalized by its maximum value, and contains the same number of contour lines. The color of the contour lines shows the relative intensity indicatedby the color bar.

resolution of the minimum-variance filter is much higher than thatof sLORETA.

where w MV (r) indicates the weight obtained from Eq. (7). The
resolution kernel is then expressed as

Discussion and conclusionl T R1b fﬃ:
Rðr; r1 Þ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
T 2
l Rb lThe weight-normalized version of the minimum-variance filter
has been used for MEG source reconstruction (Sekihara et al.,2001). The weight vector for this type of filter is given byw ðr Þ ¼

wMV ðrÞ
R1b l ðr Þ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ;
twMV ðrÞt
Tl ðrÞR2 l ðrÞ
b

ð31Þ

ð32Þ

Using Eqs. (37) and (38), the condition for the resolution kernel
having maximum at the source location is given bycosðl; f Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ b 1:
1 þ aða þ 2Þð1  cos2 ðl; f ÞÞ

ð33Þ

K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

Because a is positive, the above condition obviously holds for any
l and f. The condition for no location bias when noise exists isexpressed as
1 þ að1  cos2 ðl; f ÞÞ
b 1:
½1 þ að1  cos2 ðl; f ÞÞ þ aða þ 1Þð1  cos2 ðl; f ÞÞ

1065

adaptive filters are not influenced by source correlation. Therefore,
investigations clarifying the complementarity of both types ofspatial filters are needed, and such investigations are currently
underway, with results to be published in the near future.

ð34Þ

It is again clear that this condition holds for any l and f, and thus
the weight-normalized minimum-variance filter does not have alocation bias even when noise exists.
The comparison between Eqs. (5) and (31) shows the similarity
between the weight-normalized minimum-norm and minimumvariance spatial filters. Actually, Eq. (5) is exactly the same as Eq.
(31) if G is replaced with Rb . Assuming that the sensor noise is thespatially-uncorrelated Gaussian noise with the variance of r 02, wehave the relationship
Z Z


Rb ¼
LðrÞLT ðr VÞ hsðr; t Þsðr ;V t Þidrdr Vþ r20 I:
ð35Þ
Therefore, if hs(r, t) s(r V, t)i = r 2d (r  r V), namely if the sources
are the uniformly distributed incoherent sources, the above
relationship reduces to Rb = G + r 02 I, and the weight-normalized
minimum-norm and minimum-variance filters are equal (Mosher etal., 2003). However, in general, these two spatial filters are very
different and the weight-normalized minimum-variance filter canobtain unbiased estimates of the source locations, while the weightnormalized minimum-norm filter cannot, as we have shown here.
In summary, we have discussed the source location bias in the
single-source reconstruction for several representative spatial filtersand found that sLORETA and the lead-field normalized minimumvariance spatial filter do not have a source location bias. Whenmeasurement noise is taken into account, the SNR determines
whether the results of the sLORETA reconstruction contains a bias
in estimated source location. On the other hand, the lead-fieldnormalized minimum-variance filter has no location bias evenwhen noise exists. We compare the spatial resolution for sLORETA
and the minimum-variance filter, and show that the minimumvariance filter provide much higher spatial resolution thansLORETA. The results of these arguments are validated by ournumerical experiments as well as the experiments using evoked
responses.
This paper compares adaptive and non-adaptive spatial filterswith respect to source location bias and spatial resolution. As far asthese performance measures are concerned, the results in this paper
support the superiority of the adaptive filters over the non-adaptive
filters. It should be mentioned that, however, to really determine
the superiority of one over the other, a thorough investigation
would be required, not only of these performance measures but
also of other performance measures including robustness to various
causes of error such as the existence of correlated sources, theeffects of sample covariance, and the violation of the low-rank
signal assumption. It should be noted that the results of ouranalysis in this paper regarding the performance of the adaptive
filters indicate their asymptotic performance, which is obtained
when the ideal covariance is used. The lead-field normalized
minimum-variance filter may show some amount of the source bias
when a sample covariance matrix is obtained from a limited
number of time points. Also, it is well known that thereconstruction results from adaptive filters can be degraded whenhighly-correlated sources exist (Sekihara et al., 2002b), while non-Acknowledgments
The authors wish to thank Dr. Isao Hashimoto and Dr. Tomiaki
Kimura for providing the somatosensory data. We also thank to Dr.
Alec Marantz and Dr. David Poeppel for providing the auditorysomatosensory data. This work has been supported by Grants-inAid from the Ministry of Education, Science, Culture and Sports inJapan (C13680948 and C16500296). This work has also been
supported by grants from the National Institutes of Health(P41RR12553-03 and R01-DC004855-01A1).
Appendix A. Mathematical preliminaries
This appendix provides mathematical preliminaries that help
readers follow the discussion in Theory. Let us assume that asingle source exists at r 1. Its moment magnitude is denoted s 1 (t)and its power is denoted r 12; r 12 = hs 1(t)2i. The lead field vector at
r 1 in the source direction is denoted f, namely, f = l(r 1). Then, themeasured data b(t) is expressed as b(t) = s 1 (t) f + n(t) where n(t)is the additive noise. The noise is assumed to be white Gaussian
noise uncorrelated among sensor channels. Thus, r 02 I =hn (t) n T(t)i, where r 02 is the power of the noise. The measurement
covariance matrix Rb is given by
Rb ¼ r20 I þ r21 f f T :ð36Þ

Its inverse is expressed as
R1
b ¼

1
r20

I

a
ffT
;
1 þ a tf t2

ð37Þ

where a = (r 12 / r 02)t f t2. For the dicussion in effects of noise on
the location bias, the square inverse is derived as!
1
ð2 þ aÞa f f T
2
Rb ¼ 4 I 
:
ð38Þ
r0
ð1 þ aÞ2 t f t2
Making use of the formula in Eq. (37), we have
 T 2!
l f
1
a2
l T R1l¼
tlt
b
1 þ a t f t2
r20
¼

tlt 2
a
1
cos2 ðl; f Þ ;
1þa
r20

and
l

T

R1
b f ¼

1
r20

 T  T !
l f f f
a
lTf
l f 
¼ 2
:
2
1þa
tf t
r0 ð1 þ aÞ
T

ð39Þ

ð40Þ

Therefore, we can derive Eq. (20).
The definition of the generalized cosine between the twocolumn vectors a 1 and a 2 with the metric C, where C is a positive
definite matrix is
 T
2
a Ca2
2
;
ð41Þ
cos ða1 ; a2 jCÞ ¼  T 1  T
a1 Ca1 a2 Ca2

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K. Sekihara et al. / NeuroImage 25 (2005) 1056–1067

orjaT1 Ca2 j:cosða1 ; a2 jCÞ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
aT1 Ca1 aT2 Ca2ð42Þ

Here, the inequalities cos2(a 1, a 2|C) V 1 and cos(a 1, a 2|C) V 1
hold, because the Swartz inequality holds for any positive definite
matrix C, that is,
 T


2
a1 Ca1 aT2 Ca2 z aT1 Ca2 :ð43Þ
When C is equal to the identity matrix I, the generalized cosine issimply written as cos2(a 1, a 2), which is equal to
 T 2
a a2
:
ð44Þ
cos2 ða1 ; a2 Þ ¼  T 1  T
a1 a1 a2 a2

Appendix B. Non-adaptive spatial filter formulations
This appendix shows how the non-adaptive spatial filter
formulations described in Adaptive and non-adaptive spatial filters
can be obtained from their well-known tomographic-reconstruction
formulations. The tomographic-reconstruction formulations
assume pixels, of which locations are denoted r 1, r 2,. . ., r N where
N is the total number of pixels. The composite lead field for all
pixel locations are, then, defined as
LN ¼ ½Lðr1 Þ; Lðr2 Þ; N ; LðrN Þ :ð45ÞUsing LN , the minimum-norm reconstruction method is expressed
as
3
3
2 T
2
ŝs ðr1 ; t Þ
L ðr 1 Þ
7
6 T
6 ŝs ðr2 ; t Þ 7

7 ¼ LT LN LT 1 bðtÞ ¼ 6 L ðr2 Þ 7 LN LT 1 bðt Þ;
6
N
N
N
5
5
4
4
v
v
T
ŝs ðrN ; t Þ
L ðr N Þ
ð46Þ
where the vector ŝ(r j , t) is the estimated three-dimensional source
vector defined as

 


T
ŝs rj ; t ¼ ŝs x rj ; t ; ŝs y rj ; t ; ŝs z rj ; t :Therefore, from Eq. (46), we obtain 
1bðt Þ:ŝs rj ; t ¼ LT rj LN LTN

NXL rj LT rj ¼ G;

where w(r j ) = G1 l (r j ) and the scalar ŝ (r j , t) is the (signed)
source magnitude. This weight vector is identical to that in Eq. (3).
Using the weight matrix for the minimum-norm filter W(r j )defined above, the original form of the weight-normalized
minimum-norm method proposed by Dale et al. (2000)3 is equal to
WT rj bðt Þ
LT rj G1 bðt Þ
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
q¼
ŝs rj ; t ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ


 
 
  : ð51Þtr WT rj W rj
tr LT rj G2 L rj
Introducing the lead field in the voxel orientation l(r j ), the scalar
formulation is derived asl T rj G1 bðtÞﬃ ¼ wT rj bðtÞ;ŝs rj ; t ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð52Þ
l T rj G2 l rj
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ




1
l T rj G2 l rj . This weight vector iswhere w rj ¼ G l rj
identical to that in Eq. (5).
Finally, we derive the spatial-filter form of sLORETA. In thismethod, the minimum-norm reconstruction results are normalized
by the square root of the resolution kernel. The resolution kernel ofthe vector minimum-norm filter in Eq. (49) is obtained as a 3 3matrix WT (r j ) L (r j ) = LT (r j ) G1 L (r j ). Thus, the formulationequivalent to Eq. (20) in Pascual-Marqui (2002) is  1=2 T 
L rj G1 bðt Þ:
ð53Þ
ŝs rj ; t ¼ LT rj G1 L rj
Substituting the lead field matrix LT (r j ) with the lead field vector
in the voxel orientation, l (r j ), in the above equation, the scalarversion of sLORETA spatial filter is obtained asl T rj G1 bðtÞTŝs rj ; t ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ð54Þ ﬃ ¼ w rj bðtÞ;T
1l rj G l rj
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
l T rj G1 l rj . This weight vector iswhere w rj ¼ G1 l rj
identical to that in Eq. (6).
References

ð47Þ

In Eq. (47), the product LN LNT is equal to the gram matrix G,
which is defined in Eq. (4), that is,
LN LTN ¼

Then, the scalar spatial-filter formulation can also be derived asŝs rj ; t ¼ l T rj G1 bðt Þ ¼ w T rj bðt Þ;
ð50Þ

ð48Þ

j¼1

where errors caused due to the pixel discretization are neglected.
Therefore, the three-dimensional current vector at r j is obtained as



ð49Þ
ŝs rj ; t ¼ LT rj G1 bðt Þ ¼ WT rj bðt Þ;
where the weight matrix W (r j ) is defined as W (r j ) = G1 L (r j ).
The above equation is the vector spatial-filter formulation of theminimum-norm reconstruction method. To derive the scalar
formulation, we assume that the orientation of each voxel isknown and define the lead field in that voxel orientation as l (r j ).

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