Computing resolution forneuromagnetic imaging systems
Kensuke Sekihara

Department of Advanced Technology in Medicine
Tokyo Medical and Dental University
1-5-45 Yushima, Bunkyo-ku, Tokyo 113-8519, Japan
Signal Analysis Inc.
Hachioji, Tokyo 192-0031, JapanAbstract
This paper proposes a novel signal-detection-theory-based definition
for the resolution of neuromagnetic imaging systems, and develops a
Monte Carlo computer simulation method to compute the resolution. Using the resolution as a performance measure, the performance of various
types of sensor hardware is assessed. The assessments include performance improvements due to the increase in the number of sensors andperformance changes due to the change in the gradiometer baseline orchange in the helmet size. We compare the performance difference between planar and axial gradiometer arrays, and also compare the performance between the conventional radial sensor array and a vector sensor
array. We compute the resolution of two existing neuromagnetic sensor arrays, MEGvisionTM (Yokogawa Electric Corporation, Tokyo, Japan) andElekta-Neuromag TRIUXTM (Elekta Corporate, Stockholm, Sweden).

1

Introduction

Electrophysiological activity of neurons in the cerebral cortex generates tinymagnetic fields outside the scalp. Direct non-invasive measurements of thisneuronal activity on a sub-millisecond time scale can be achieved by magnetoencephalography (MEG)[1][2]. Modern MEG systems are capable of wholehead coverage with simultaneous measurements by nearly 300 sensors. Suchwhole-head sensor arrays, together with advanced signal processing algorithms,
now enable imaging of dynamic brain activity – referred to as neuromagnetic
imaging[3][4][5].
One problem with neuromagnetic imaging is that no clear measure exists
for assessing the overall performance of imaging systems. A naive (but very
popular) index used for assessing the performance of neuromagnetic imaging
systems is the number of sensors. Although one can imagine that a system withlarger number of sensors should have better performance than that of a systemwith smaller number of sensors, it has been difficult to quantitatively assess theperformance improvements due to increasing the number of sensors.
Theoretical assessments of the performance of MEG multi-channel sensor
systems have been proposed that apply an information theoretic approach to thesensor data[6]. It may be true that a sensor array that captures a greater amountof information gives higher-quality source images. However, since a quantitative
relationship between the information content captured in sensor data and thequality of source images is unknown, such an information theoretic analysis
on the sensor data is insufficient for quantitative performance assessments ofneuromagnetic imaging systems.
Resolution expresses a system’s capability to discriminate two closely locatedsources and has conventionally been used to assess the performance of a wide
variety of imaging systems, from classic optical instruments to medical imaging
systems including X-ray computed tomography(CT) and magnetic resonanceimaging(MRI). However, it is difficult to define the resolution of neuromagnetic

2

imaging systems because this capability depends on the location and orientationof sources because of the space-variant nature of this imaging method.
Therefore, to assess the resolution of neuromagnetic imaging systems, some
type of Monte Carlo-based method is needed in which the source locations andorientations are randomly chosen in each Monte Carlo trial. This paper develops
such a Monte Carlo-based method, which incorporates a novel definition ofresolution computed based on the results from Monte Carlo experiments.
Using the resolution, we can explore the relationship between the system’s
performance and choices of various hardware design parameters. We can make aquantitative comparison among existing sensor systems and predict the performance of future systems. In Section 2, we describe, in detail, how the resolution
for neuromagnetic imaging systems is defined and computed. Using the resolution as a performance measure, we compare various types of sensor hardware in
Section 3. We also compare the resolution of two existing neuromagnetic imaging systems, MEGvisionTM (Yokogawa Electric Corporation, Tokyo, Japan) andElekta-Neuromag TRIUXTM (Elekta Corporate, Stockholm, Sweden).

2Monte Carlo method for computing spatial
resolution

2.1

Sensor data generation

We have developed a Monte Carlo simulation method to compute the resolutionof neuromagnetic imaging systems. In this method, we assume two sources thathave equal intensity. The locations and orientations of the two sources are
randomly chosen in each Monte Carlo trial, with the distance of the sources∆ fixed across all trials. The locations of the two sources are confined within
the simulated brain region shown in Fig. 1. A spherical-shell region with outerradius 8cm and inner radius 1cm is defined as the brain region and the region
more than 4cm below the sphere center is excluded. The center of the brain
region is 10.5 cm below the sensor located at the center of the sensor array.
The two sources have random time courses generated using Gaussian random
numbers; the time courses have 4000 time points. One hundred Monte Carlotrials of simulated sensor data are generated, each with random source locations
and orientations with a fixed inter-source distance. Gaussian white noise isadded to this signal magnetic field to generate the simulated sensor data. The
standard deviation of the sensor noise is fixed to 50 fT throughout the computer
simulation in this study1 . A two-orientation forward lead field is calculated using
the single spherical-shell model[7], and to compute the signal magnetic field, thesource activity is projected to the sensor space through the lead field.
1 The rationale for this 50 fT value is as follows: Modern SQUID-based biomagnetometers√
typically have the noise level of 5 fT/ Hz, so if we assume the bandwidth of the measurement
to be 100 Hz, then the measured data contain a noise level of 50 fT.

3

2.2

Source reconstruction and local peak detection

The simulated brain volume in Fig. 1 is segmented into voxels. In our computer
simulation, the voxel size V is determined according to the inter-source distance,such thatV = 0.2∆∆ ≤ 2.5 cm,(1)

V = 0.5 cm∆ > 2.5 cm.

(2)

The rationale for determining the voxel size in this manner is given in the
Appendix. Adaptive beamformer source reconstruction[4][8] is performed and asource power map (the power of the time courses at each voxel) is computed at
all voxel locations. The localization accuracy is then assessed in the following
manner.
If a voxel has a value higher than its nearest neighbors in three dimensions,
we determine that a “local peak” exists at that voxel. After all the local peaklocations are searched, we test whether each local peak is within a particulardistance D from one of the true source locations. If a particular peak is within
D from a true source location, that peak is labeled as a “detected source”. If
there is no true source within D from the local peak, that local peak is labeled
a “false detection”. In our computer simulation, the distance D is referred toas the detection radius. Using the voxel size V , the detection radius is set suchthat
D = 1.5V.
(3)
The rationale for determining D in this way is presented in the Appendix.

2.3

A-prime metric

Capability for detecting sources is assessed using a signal-detection-theory-based
metric called the A-prime metric[9]. To compute the A-prime metric, we firstcompute the hit rate H, which is a ratio describing the correct source detections.
We define the numbers of detected sources nD and undetected sources nU . Using
the relationship nD + nU = 2 where 2 is the total number of true sources, thehit rate H is computed by
H=h

nD
hnD i
i=
,
nD + nU2(4)

where h·i indicates averaging across all Monte Carlo trials.
Defining the number of false detections in each Monte Carlo trial as nF , wecompute the false-detection rate F as follows:
nF
nF
F =h
i=h
i,
(5)
nD + nF
nL
where nL is the number of total local peaks in each Monte Carlo trial, andthe relationship nL = nD + nF is used2 . Once H and F are obtained, we can
2 The definition of the false-detection rate above differs from that used in [10], where F is
obtained using F = nF /(maximum number of nF ) and the resultant F becomes significantly
different than that calculated using Eq. (5) when the signal-to-ratio (SNR) is low.

4

compute the A-prime metric, Ap , which is equal to the area under the receiver
operating characteristics (ROC) curve for H versus F . Since we have only asingle pair of H and F , the first-order approximation is given by
Ap =

H −F
1
+ .
2
2

(6)

The above equation is derived in the Appendix. It should be noted that theA-prime metric was used to evaluate the performance of source reconstruction
algorithms in previous studies[10][11][12].

2.4

Spatial resolution

The spatial resolution is generally defined as the minimum distance at which thetwo sources are separately detected. In our Monte Carlo computer simulation,
the A-prime metric is obtained for two sources with a fixed inter-source distance.
We repeat the Monte Carlo simulation with different inter-source distances andplot the A-prime metric versus the inter-source distance.
We then look for the source distance that gives the A-prime metric equal toa certain value α (0.5 ≤ α ≤ 1), and define this source distance to be the spatial
resolution of the imaging system for a given source intensity. By repeating thewhole procedure with various source intensities, we obtain a plot of resolution
versus source intensity, which represents the resolution of the imaging system
under assessment.

2.5

Error-bar estimation using bootstrap method

To obtain the confidence interval of the spatial resolution, we use the bootstrap
method. Denoting the number of Monte Carlo trials as M , we have M resultsM1
for nD and nF , which are denoted n1D , . . . , nM
D and nF , . . . , nF . In the bootstrap
1
method, we first choose M bootstrap samples from nD , . . . , nM
D with replace.
Wethen computeeMments, and the bootstrap samples are denoted n
e1D , . . . , nDe We also compute the bootstrap false detection rate Fe
the bootstrap hit rate H.
M
1using bootstrap samples n
e1F , . . . , neM
F taken from nF , . . . , nF with replacements.
e and Fe, and we plot
We can now compute the bootstrap A-prime using H
the bootstrap A-prime versus the inter-source distance. The resolution, which is
the inter-source distance that gives the A-prime equal to α, is finally obtained.
Since there are many ways to choose the bootstrap samples, we can obtain
the bootstrap distribution of the estimated resolution and finally obtain theconfidence interval of the estimated resolution values by using the bootstrap
distribution. We set error bars to the 95% confidence interval in this paper.

5

3

Results of computing resolution for various
types of neuromagnetic imaging systems

3.1

Resolution computation

We assume four types of sensor arrays having 80, 160, 320, and 640 sensors forresolution assessment experiments. In these sensor arrays, sensors are aligned
on a surface of a spherical helmet that has a 13 cm radius. The sensor locationsof these four types of sensor arrays are shown in Fig. 2. As shown here, threequarters of the helmet surface are covered by sensors, which are arranged withan equal inter-sensor spacing. In our experiments, the sensor is a first-orderaxial gradiometer with a 5 cm baseline unless otherwise noted.
We perform the Monte Carlo computer simulation described in the preceding
section. Two sets of source reconstruction results are shown as examples inFig. 3. For these results, the two sources are 3 cm apart, and the sensor systemwith 160 sensors is assumed. Here, Fig. 3(a) shows a fairly high signal-to-noise
ratio (SNR)3 case (SNR of 2) in which the two sources are detected. A low
SNR case (SNR of 0.5) in which the two sources are not resolved is shown in
Fig. 3(b). Obviously, the A-prime metric is equal to one in the first case. It isequal to zero in the second case since the hit rate is zero and the false detection
rate is one.
By performing the Monte Carlo experiments with various inter-source distances, the dependence of the A-prime metric on the inter-source distance isobtained. Examples of plotting the three metrics, namely the hit rate, the false
detection rate, and the A-prime metric, with respect to the inter-source distanceare shown in Fig. 4. The results in Fig. 4(a) are obtained using the source intensity set to 60 nAm, which gives an average SNR of 3. In this figure, we can see aclear tendency that the hit rate increases, the false-detection rate decreases, andthe resultant A-prime metric increases as the inter-source distance increases.
In our computer simulation, the resolution is defined as the inter-source
distance that gives an A-prime metric of 0.75. In Fig. 4, the level of Ap = 0.75
is shown by a horizontal broken line, and the vertical broken line indicates theresolution, which is the inter-source distance giving Ap = 0.75. The plot in
Fig. 4(a) shows that the resolution in this case is slightly larger than 2 cm and,using the interpolation, the resolution is computed to be 2.1 cm.
The results in Fig. 4(b) show an example for a case where the resolution is
undefined because the value of A-prime never reaches 0.75, even when the intersource distance is as large as 6 cm. These plots were obtained with a source
intensity of 6 nAm, which gives an average SNR of 0.3.
Although the adaptive beamformer algorithm is used as the source reconstruction algorithm throughout this study, we show the results of attempting
to compute the resolution using the sLORETA algorithm[13] for comparison.
3 Throughout this paper, the SNR is defined as the ratio between the square root of thepower of the generated signal magnetic field and the square root of the power of the noise,
which is 50 fT.

6

Plots of the hit rate, the false-detection rate, and A-prime with respect to theinter-source distance, obtained using sLORETA, are shown in Fig. 5. Here, thesource intensity is set to 400 nAm, which corresponds to an SNR of 20. This
figure shows that, even when the SNR is very high, the A-prime plot neverreaches 0.75, and the spatial resolution is not defined under the definition usedin this paper.
By repeating the Monte Carlo experiments with different source intensities,
a plot of resolution with respect to source intensity can be obtained. Such a plot
obtained with an array of 160 sensors is shown in Fig. 6. Here, the relationshipbetween mean SNR and source intensity is shown in the lower panel. The
source intensity is varied from 4 to 400 nAm. This range corresponds to the
SNR range of approximately 0.2 to 20, which is the range encountered in most of
MEG measurements. In Fig. 6, we can see that the resolution is approximately
3 cm for this sensor system when the source intensity is 20 nAm, which gives
an SNR of 1. At a very high SNR ratio such as 20, the resolution becomes as
small as 1 cm.
Figure 6 indicates that, for the source intensity range between 10 and
100 nAm, the plot of resolution can be regressed by a linear regression model:yb(I) = a(log10 I) + b,

(7)

where I indicates the source intensity, and yb(I) indicates the modeled resolution(cm) at the source intensity I. The regression coefficients are obtained by leastsquares fitting to data between 10 and 100 nAm, giving a = −3.0, and b = 7.4.
The regression results are shown by the broken line in Fig. 6. The results
indicate that increasing SNR by an order of magnitude improves resolution by
3 cm. In other words, a 3-cm improvement in resolution arises from a 10-foldimprovement in SNR.
Note that the linear regression mentioned above was performed for the data
in range of 10 and 100 nAm. The source intensity range from 10 to 100 nAm
corresponds to the SNR range between 0.5 and 5 according to the bottom panelin Fig. 6. This range of SNR is most often encountered in MEG measurements,
and thus it is referred to as the practical SNR range in this paper.

3.2

Dependence of resolution on the number of sensors

We next investigate how the resolution of neuromagnetic imaging systems depends on the number of sensors. Plots of resolution for the four types of sensor
arrays in Fig. 2 are shown in Fig. 7(a). The plots in Fig. 7(a) show a generaltendency toward improved resolution when the number of sensors is increased.
Although the difference in resolution becomes small under high SNR conditions,
there are clear improvements in resolution for the source intensity from 10 to
100 nAm, which corresponds to the SNR range between 0.5 and 5, the practicalrange of SNR.
To derive a clear quantitative relationship between resolution and numberof sensors, we remove the dependence of resolution on source intensity. To do

7

so, the resolution is standardized by that of the 160-sensor array. Let us define
the standardized resolution of a k-sensor system at the source intensity I asx(k) (I). The value of x(k) (I) is computed using x(k) (I) = y(k) (I)/y(k=160) (I),where y(k) (I) is the resolution of a k-sensor system at source intensity I. We
then apply the linear regression analysis to the standardized resolution datax(k) (I). That is, we use:xb(k) (I) = a(k)(log10 I) + b(k),(8)

where xb(k) (I) is the modeled standardized resolution computed using the above
equation.
The results of the linear regression are shown by broken lines in Fig. 7(b).
Here, we see that the dependence of the standardized resolution on source intensity becomes very small but the dependence is not completely removed. We
thus further average the standardized resolution over the practical SNR range,such thatβ(k) = hb
x(k) (I)i10≤I≤100 ,(9)where h·iA indicates the average over the region I ∈ A. This mean value of β(k),referred to as the effective standardized resolution, is considered to represent thestandardized resolution for a sensor system under assessment.
The plot of the effective standardized resolution, β(k), with respect to thenumber of sensors k is shown in Fig. 8. The plot shows that there is a relationship between log k and β(k) that is approximately linear when k is less than300. However, when k becomes greater than 300, the plot starts to deviate fromthe linear relationship.
We therefore apply a second-order regression model to this plot and theregression results are shown by the broken line in Fig. 8; the model fitted to theplot is expressed asbβ(k)= 0.37(log10 k)2 − 2.2 log10 k + 4.05,(10)bwhere β(k)is the modeled effective standardized resolution. The resolution fora given number of sensors can be predicted using Eq. (10). For example, thisbbequation gives β(64)= 1.28 and β(320)= 0.86, so the increase in the numberof sensors from 64 to 320 attains a 50% resolution improvements. However, itbbalso gives β(640)= 0.785 and β(1000)= 0.774. The results indicate that, evenif we increase the number of sensors from 640 to 1000, the improvement in the
resolution is only 1%.

3.3

Resolution dependence on the helmet radius

The analysis described so far assumes that sensors are arranged on a surface
of a spherical helmet with a 13 cm radius. Existing SQUID-based whole-head
sensor systems have values more or less similar to 13 cm for the radius of their
helmets, although their helmets are not exactly spherical. Requirements for the
thickness of the helium dewar wall may result in similar helmet sizes. However,8due to the recent development of non-SQUID-type room temperature sensors,
it may be possible to develop a whole-head sensor system with a helmet having
a much smaller radius. Therefore, the dependence of resolution on helmet sizeis next investigated.
We compute the resolution of arrays with 80, 160, 320, and 640 sensors forhelmet radii of 11.5 cm and 10 cm. (The resolution for the helmet radius of
13 cm was computed, and the results are shown in Fig. 7.) In Fig. 9, the plots
for the 160-sensor system are selectively shown. A general tendency for a smaller
helmet system to give better resolution can clearly be seen in this figure.
The resolution values are then standardized by using the values of the 160sensor system with the helmet size of 13 cm. That is, defining the raw resolution
value of the k-sensor system and the helmet size of R as y(k,R) (I), the standardized resolution is computed using
x(k,R) (I) = y(k,R) (I)/y(k=160,R=13) (I).

(11)

The same linear regression analysis as described in the preceding subsection
is applied to x(k,R) (I) to derive the modeled standardized resolution xb(k,R) (I).
Figure 10 shows the raw resolution plots for 80-, 160-, 320-, and 640-sensor
arrays in the top panels. The bottom panels show the results of the modeled
standardized resolution analysis.
We next derive the effective standardized resolution β(k, R) by using
β(k, R) = hb
x(k,R) (I)i10≤I≤100 . The effective standardized resolution is plottedwith respect to the number of sensors in Fig. 11. Here, the effective standardized resolution for the each of the three helmet sizes is plotted. The
observation that these three plots are nearly parallel suggests that the resolution improvement due to the use of a smaller helmet is nearly independent from the number of sensors. To check this point, the effective resolution
β(k, R) is further normalized by the corresponding value when the helmet size
is 13 cm. That is, the normalized effective resolution, β̄(k, R), is obtained byusing β̄(k, R) = β(k, R)/β(k, R = 13), and the results are shown in Table I.
The results in Table I were analyzed by two-way analysis of variance (2DANOVA) with the level of significance set to 0.005. The analysis indicatesthat the dependence of β̄(k, R) on the number of sensors k is not statisticallysignificant but the dependence on helmet radius R is statistically significant.
Therefore, the mean values obtained by averaging β̄(k, R) over k should represent the dependence of resolution on helmet size. Table I shows that, comparedto the case of the 13-cm helmet, nearly 20% and 35% resolution increases arerespectively attained if we use sensor helmets with radii of 11.5 cm and 10 cm.
Table I Normalized effective resolution β̄(k, R).
helmet radius R (cm)
13
11.5
10

number of sensors, k
80
160 320 640
11
1
10.83 0.79 0.85 0.78
0.67 0.63 0.65 0.64

9

mean
1
0.81
0.65

3.4

Resolution dependence on gradiometer baseline

We assess the effects of gradiometer baseline on resolution. Note that an axial
gradiometer with a baseline of 5 cm has been assumed in our analysis presented
so far. We here compute the resolution of 80-, 160-, 320-, and 640-sensor systems
with axial gradiometers baseline B of 1.6 cm, 3.2 cm, 5.0 cm, and ∞4 cm.
Results are shown in Fig. 12. Here, the helmet radius is fixed at 13 cm. In
this figure, we can observe a general tendency for a sensor system with a larger
baseline to give a better resolution.
To make quantitative assessments on this tendency, we analyze the effective
standardized resolution, i.e., we standardize each resolution value by using thecorresponding resolution value when the baseline is 5 cm, and apply the linear
regression analysis to obtain the effective standardized resolution. The results
are shown in Table II.
We analyzed the results in Table II by 2D-ANOVA with the level of significance set to 0.005. The analysis indicates that the dependence on numberof sensors k is not statistically significant but the dependence on the baseline
value B is statistically significant. The mean values of the effective resolution
across k thus show the dependence of the resolution on the baseline values. According to Table II, it can be seen that, compared to the 5 cm baseline case,
34% and 11% resolution losses occur if we use gradiometers with 1.6 cm and3.2 cm baselines, respectively. On the other hand, use of a gradiometer with
5 cm baseline causes only 10% loss of spatial resolution, compared to the use ofa magnetometer sensor.
Table II Effective standardized resolution for four valuesof gradiometer baseline.

number of sensors
80
160
320
640
mean

3.5

1.6
1.28
1.37
1.33
1.39
1.34

baseline (cm)3.2
5∞1.06 1.00 0.84
1.14 1.00 0.91
1.12 1.00 0.96
1.09 1.00 0.91
1.10 1.00 0.90

comparison between planar and axial gradiometer arrays

We here assess the resolution of planar gradiometer arrays, and compare theresolution with that of axial gradiometer arrays. To do so, we define threeorthogonal directions (e, f , g) as shown in Fig. 13. The radial direction is definedas the e direction, and the two tangential directions are defined as the f and g
directions in which f and g indicate the longitudinal and latitudinal directions,
respectively.
4 A gradiometer with B = ∞ indicates the magnetometer.

10The planar gradiometer in which the two sensors are aligned in the longitudinal direction is referred to as the f -gradiometer, and the planar gradiometer inwhich the two sensors are aligned in the latitudinal direction is referred to as theg-gradiometer. We also assess the resolution of an “orthogonal gradiometer”,
which has a pair of the f and g gradiometers at the same location. (Note that theorthogonal gradiometer configuration is used in Elekta-Neuromag TRIUXTM .
) Since a planar gradiometer with a large baseline is physically infeasible, we
assess planer gradiometers with a 1.6 cm baseline and with a 3.2 cm baseline.
We compute the resolution of the f -gradiometer array, g-gradiometer array,and orthogonal gradiometer array. The sensor locations are the same as thoseof the axial 160-sensor array with a 13 cm helmet radius. Thus, the number of
sensors is 160 for the f - and g-gradiometer arrays, and 320 for the orthogonal
gradiometer array. The results of computing resolution are shown in Fig. 14.
The results for the 1.6 cm baseline case are shown in the top panel and thosefor the 3.2 cm baseline case are shown in the bottom panel. The results for the
160 and 320 axial gradiometer arrays with the same baselines are also shownfor comparison.
The results in Fig. 14 show that, although the axial gradiometer arrays haveresolution significantly better than that of the f -gradiometer alone array andg-gradiometer alone array, the resolution for the orthogonal gradiometer arrayis better than that of the axial 160 gradiometer array. However, for a faircomparison, the orthogonal gradiometer array should be compared to the axial
320 gradiometer array, whose results are shown by the broken lines. It can beseen that the resolution of the orthogonal gradiometer array is very close to thatof the 320 axial gradiometer array, and this indicates that the performance ofthe planar and the axial gradiometer arrays with the same baseline is nearly
equal when we use the orthogonal gradiometer configuration.

3.6

Effects of vector field measurements on resolution

We here assess the effects of vector field measurements on resolution. The directions defined by the vectors (e, f , g) are again used, and sensors that measuresthe magnetic field in the f and g directions are respectively referred to as thef - and g-sensors. The sensors that measure the magnetic field in the radial (e)direction are referred to as the radial sensors, which have been considered in
the preceding subsections. The sensors that measure the magnetic field in the
all three (e, f , g) directions are referred to as the vector sensors.
We compute resolution of four types of sensor arrays: the radial sensor array,the f sensor array, the g sensor array, and the vector sensor array. Here, alltypes of sensors are assumed to be magnetometers, and the sensors are arranged
on a spherical helmet with a 13 cm radius. The sensor locations are the same as
those of the 160 sensor array, and thus the number of sensors is 160 for all typesof sensor arrays except the vector sensor array, which has 480 (160×3) sensors.
The resolution plots for these four types of sensor arrays are shown in Fig. 15.
It can be seen in this figure that the resolution of the radial sensor array is significantly better than that of the two types of tangential sensor arrays. Although11the vector sensor array attains the highest resolution, the difference between
the vector and the radial sensor arrays is small.
To quantitatively assess the resolution differences among these sensor arrays,
we apply the standardized resolution analysis where the resolution is standardized using the corresponding values of the radial sensor array. The results ofcomputing the modeled standardized resolution are shown in Fig. 16(a). We
then compute the effective resolution by averaging the modeled standardized
resolution between I=10 and 100 nAm. Let us define the effective resolution
for the radial, f -sensor, g-sensor, and vector sensor arrays as βrad , βf , βg , andβvec , respectively. The results of computing these effective resolution are givenin the first line of Table 3. These results indicate that, compared to the useof the radial sensor array, an 80% and 40% resolution losses arise when the f and g-sensor arrays are used. The use of the vector sensor array can attain 15%
better resolution.
Table III Results of computing effective standardized resolution for radial,tangential, and vector sensor arrays5

sensor arrangement
type I
type II

type of sensors
βrad βfβg βvec1
1.8 1.4 0.851
2.4 1.8 0.91

The computer simulation described so far uses the assumption that the radial
and tangential sensors are aligned on the same sphere surface with a 13 cm
radius. This assumption is somewhat unrealistic for the tangential sensors.
Typical sensors in existing MEG arrays are coils with a 2–3 cm diameter. If weuse such coils for the tangential sensors, the center of the coil should be arranged
1–1.5 cm above the surface of the helmet. Namely, the tangential sensor arrayshould use a helmet with a radius at least 1 cm larger than the radius of thehelmet used for the radial sensor array. This fact should be taken into accountwhen we compare performance between the radial and tangential sensor arrays.
We compute the resolution assuming that the tangential sensors are arranged
on a sphere with a radius of 14 cm. The results of computing the modeled
standardized resolution in this case are shown in Fig. 16(b), and the results ofcomputing the effective resolution are given in the second line of Table III. These
results indicate that the resolution difference between the radial and tangential
sensor arrays is as large as 80% for the g-sensor array and 140% for the f -sensor
array. The resolution improvements due to the use of the vector sensor array isless than 10%, despite the vector sensor array uses 3 times more sensors suchas 480 sensors in the case of the study described here.
5 In this table, Type I sensor arrangement indicates that all types of sensors are arranged
on a sphere with a 13 cm radius. Type II sensor arrangement indicates that the radial sensors
are arranged on a sphere with a 13 cm radius but the tangential sensors are arranged on asphere with a 14 cm radius.

12

3.7

Resolution of existing sensor arrays

We compute the resolution of two types of existing neuromagnetic imaging
systems: MEGvisionTM (Yokogawa Electric Corporation, Tokyo, Japan) andElekta-Neuromag TRIUXTM . MEGvisionTM has 160 axial gradiometers with a
5 cm baseline[14], and Elekta-Neuromag TRIUXTM has a total of 306 sensors
consisting of 102 orthogonal planer gradiometers and 102 magnetometers[15].
Both systems have sensors arranged on a helmet-shaped surface with a radiusof approximately 13 cm, although their helmets are not exactly spherical.
The results of computing resolution-source-intensity-SNR plots are shown in
Fig. 17. Here, to compute SNR, the sensor noise with a standard deviation of
50 fT is again assumed. Here, the plot labeled“Y” indicates the results from
MEGvisionTM . The plot labeled“El-G” indicates the results from TRIUX in
which only the planar gradiometer sensors are used, and the plot labeled “ElG+M” indicates the plot from TRIUX in which both the planar gradiometerand magnetometer sensors are used. The range of the source intensity from 4
to 200 nAm corresponds to the SNR range from 0.3 to 15 for MEGvisionTM
and 0.2 to 10 for Elekta-Neuromag TRIUXTM . The results in Fig. 17 show thatthe resolution of MEGvisionTM is almost the same as that of Elekta-Neuromag
TRIUXTM . However, if only the planar gradiometers are used, the resolutionof TRIUXTM is significantly lower than that of MEGvisionTM in the practical
SNR range.

4

Conclusions

This paper proposes a novel signal-detection-theory-based definition for the resolution of neuromagnetic imaging systems, and develops a Monte Carlo computer simulation method to compute the resolution. Using the resolution as aperformance measure, various types of sensor hardware are compared. We first
analyze the performance changes due to change in the number of sensors, andderive an empirical equation that expresses the relationship between resolution
and number of sensors. We then analyze the influence of the sensor helmet sizeon resolution, and find that sensor helmets with radii of 11.5 cm and 10 cm,
respectively, provide nearly 20% and 35% better resolution than the helmet
with a 13 cm radius. We analyze the performance changes due to changes ingradiometer baseline and find that, compared with the 5 cm baseline case, 34%
and 11% resolution losses are respectively caused if we use a gradiometer witha 1.6 cm or a 3.2 cm baselines. On the other hand, use of a gradiometer witha 5 cm baseline causes 10% loss of spatial resolution, compared to the use of amagnetometer sensor. We find that the planar and axial gradiometer systems
attain almost the same resolution. We also compare performances between theradial and vector sensors, and find that, the resolution improvement due to theuse of the vector sensor array is only 10–15 %. Finally, we compute the resolution of two markedly different existing MEG systems, MEGvisionTM (Yokogawa
Electric Corporation, Tokyo, Japan) and Elekta-Neuromag TRIUXTM , (Elekta

13

Corporate, Stockholm, Sweden), and find that these two systems have almost
the same resolution.

5

Acknowledgements

This work was supported by a Grant-in-Aid from the Japanese Ministry of
Education, Culture, Sports, Science, and Technology (No 26282149) and by agrant from Konika-Minorta Corporation. The author is grateful to Ei HIyama
and Ryo Koga for their dedication in data analysis.

A

Appendix

A.1

Determination of voxel size

The voxel size V must be determined such that the voxel discretization does notcause additional inaccuracy in source detection. To explore how to determine
V , let us consider four scenarios with different ∆ and V settings where ∆ is theinter-source distance. The scenarios are depicted as a one-dimensional modelin Fig. 18(a)–(d) . Here, the locations of the two sources are indicated by filledsquares, and voxels are indicated by filled circles. The voxels neighboring thesources are labeled as “A”, “B”, “C”, and “D”. Here, we assume that an ideal
reconstruction is performed with no source localization bias and no blur, thatis, only the voxel nearest a source location has a reconstructed source power(indicated by a vertical arrow) due to that source. Other voxels have no source
power and their voxel values are zero.
In Fig. 18(a), a case where V is greater than ∆/2 is shown. In this case, twovoxels can exist between the two sources, such voxels are labeled“B” and “C”
in the figure. Voxel “B” is the voxel nearest Source 1 and has a reconstructed
power due to that source. Also, voxel “C” is the voxel nearest Source 2 and has
a reconstructed power due to that source. As a result, there is only a single local
peak either at voxel“B” or at voxel “C” and the two sources are not resolved in
the reconstructed image.
Figure 18(b) shows a case where the voxel size V is equal to ∆/2. In thisfigure, the voxel nearest Source 1 is voxel “A”, which has a power due to thissource. The voxel nearest Source 2 is voxel “C”, which has a power due to thissource. As a result, two local peaks exist at voxels“A” and “C”, and the twosources are resolved. However, there is a “worst case scenario”, which is shownin Figure 18(c). In this scenario, the two sources are exactly at the midpoint of
two voxels. That is, Source 1 is located at the midpoint between voxel “A” andvoxel “B”, and it is equally possible that voxel “A” has a reconstructed source
power or that voxel “B” has the reconstructed power. Figure 18(c) shows a casewhere voxel “B” has the power from Source 1 and voxel “C has the power from
Source 2. In such a case, there is a single local peak either at voxel “B” or atvoxel “C”, and the two sources are not resolved.

14

Finally, a case where the voxel size is smaller than ∆/2 is shown in Figure 18(d), in which Source 2 is still located at the midpoint between voxels “C”
and “D” and the voxel nearest Source 1 is voxel “A”. Therefore, we have local
peaks at voxel “A” and voxel “C”, and the two sources are resolved. The onedimensional analysis described above indicates that, to avoid voxel discretization
errors in local peak detection, the voxel size V should be smaller than half of
the inter-source distance ∆.
The arguments above are extended to three-dimensions. A three-dimensionalsource-and-voxel configuration, corresponding to the one-dimensional case in
Fig. 18(c), is shown in Fig. 18(e). In this figure, the two sources are locatedat the center of the cubic grid of voxels, and an empty cube exists between thetwo cubes containing the√ sources. The inter-source distance between the twosources ∆ is equal to 2 3V . The above arguments for the one-dimensional caseindicate that the voxel size V should be determined to satisfy∆
V < √ = 0.28∆.
2 3

(12)

In our Monte Carlo simulation, the voxel size is set to 0.2∆ when ∆ ≤ 2.5 cm,
and it is set to 0.5 cm when ∆ > 2.5 cm.

A.2

Determination of detection radius D

We next determine the detection radius D. Figure 19(a) shows the threedimensional voxel cubic grid with a source located inside the cube. Under
the ideal assumption that the reconstruction has no localization bias and no
blur, the voxel closest to the source has a reconstructed source power. If thesource exists at exactly the center of the cube, the eight voxels at the vertices
have an equal probability of having a reconstructed power. In this case, thedistance between
the source and one of the voxels, denoted X in Fig. 19(a),
√
is
equal
to
3V
/2.
Therefore,
the detection radius D must be greater than√√
3V /2, because if D < 3V /2 the reconstructed source may not be counted asa “detected source“, even when the ideal reconstruction is performed.
We next determine the upper limit of D. If we set a large value to D,
wrong reconstruction may be counted as a correct detection. To avoid suchpossibilities, D should be determined so as not to include voxels belonging to
adjacent cubes. In Fig. 19(b), two neighboring cubic grids of voxels are shown
and a source is located at the center of the lower cube. In this figure, the voxel
nearest the source but belonging to the upper cube is marked by a triangle.
The√ distance between this voxel and the source is denoted Y , which is equal to
11V /2. Therefore, to exclude this voxel, D should satisfy the relationship,√√3
11
V ≈ 0.87V < D <V ≈ 1.66V.
(13)
2
2
In our Monte Carlo simulation, we use the value D = 1.5V for the detection
radius.
15

A.3

Derivation of Eq. (6)A-prime is defined as the area under the ROC curve[9]. The first-order approximation of this area is the area of a rectangle whose corners have coordinates
defined by (0, 0), (F, H), (1, 1), and (1, 0), as shown in Fig. 20. Thus, thefirst-order approximation of the A-prime metric is expressed asAp =HF + (H + 1)(1 − F )
H −F
1
=
+ ,
2
2
2

(14)

which is Eq. (6).

References[1] M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa, “Magnetoencephalography-theory, instrumentation, and applications
to noninvasive studies of the working human brain,” Rev. Mod. Phys.,vol. 65, pp. 413–497, 1993.
[2] S. Supek and C. J. Aine, Magnetoencephalography: From Signals to Dynamic Cortical Networks. Springer, 2014.
[3] K. Sekihara and S. S. Nagarajan, Electromagnetic Brain Imaging: A
Bayesian Perspective. Springer, 2015.
[4] K. Sekihara and S. S. Nagarajan, Adaptive spatial filters for electromagnetic
brain imaging. Berlin, Heidelber: Springer-Verlag, 2008.
[5] S. Baillet, J. C. Mosher, and R. M. Leahy, “Electromagnetic brain mapping,” IEEE Signal Processing Magazine, vol. 18, pp. 14–30, 2001.
[6] J. Nenonen, S. Taulu, M. Kajola, and A. Ahonen, “Total information
extracted from MEG measurements,” in International Congress Series,
vol. 1300, pp. 245–248, Elsevier, 2007.
[7] J. Sarvas, “Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem,” Phys. Med. Biol., vol. 32, pp. 11–22, 1987.
[8] B. D. Van Veen, W. Van Drongelen, M. Yuchtman, and A. Suzuki, “Localization of brain electrical activity via linearly constrained minimum variance spatial filtering,” IEEE Trans. Biomed. Eng., vol. 44, pp. 867–880,
1997.
[9] J. Snodgrass and J. Corwin, “Pragmatics of measuring recognition memory:
applications to dementia and amnesia,” J. Exp. Psychol. Gen., vol. 117,
pp. 34–50, 1988.
[10] J. P. Owen, D. P. Wipf, H. T. Attias, K. Sekihara, and S. S. Nagarajan,“Performance evaluation of the champagne source reconstruction algorithm
on simulated and real M/EEG data.,” NeuroImage, vol. 60, pp. 305–23,
2012.
16

[11] D. P. Wipf, J. P. Owen, H. T. Attias, K. Sekihara, and S. S. Nagarajan,“Robust Bayesian estimation of the location, orientation, and time courseof multiple correlated neural sources using MEG,” NeuroImage, vol. 49,
pp. 641–655, 2010.
[12] J. M. Zumer, H. T. Attias, K. Sekihara, and S. S. Nagarajan, “A probabilistic algorithm integrating source localization and noise suppression for
MEG and EEG data,” NeuroImage, vol. 37, pp. 102–15, 2007.
[13] R. D. Pascual-Marqui, “Standardized low resolution brain electromagnetic
tomography (sLORETA): technical details,” Methods and Findings in Experimental and Clinical Pharmacology, vol. 24, pp. 5–12, 2002.
[14] G. Uehara, Y. Adachi, J. Kawai, M. Shimogawara, M. Higuchi, Y. Haruta,
H. Ogata, and H. Kado, “Multi-channel SQUID systems for biomagnetic
measurement,” IEICE Trans. on Electronics, vol. E86-C, pp. 43–54, 2003.
[15] “http://www.elekta.com/healthcare-professionals/products/elektaneuroscience/functional-mapping/elekta-neuromag-productspecifications.html.”

17

8cm

10.5cm1cm
4cm

Figure 1: Simulated brain region assumed for Monte Carlo computer simulation.
The center of the brain region is 10.5 cm below the sensor located at the centerof the sensor array. A spherical-shell region with the outer radius of 8 cm andthe inner radius of 1cm is defined as the brain region where the region morethan 4 cm below the sphere center is excluded.

18

15

15
10

10

5
0

cm

cm

5
0

-5
-5

-10

-10
-20

-15
-20
0
20

-20

-10

10

0

20

15

15

10

10

5

5

0

20

cm

cm

cm

cm

cm

0

-10

0

10

20

0

10

20

cm

0

-5

-5

-10

-10

-15
-20

-20

-15
-20
0
20
cm

-20

0

-10

10

20

0
20
cm

cm

-20

-10
cm

Figure 2: Four types of virtual whole-head sensor arrays with 80 sensors (topleft), 160 sensors (top right), 320 sensors (bottom left), and 640 sensors (bottom
right). The filled circles show the sensor locations. Sensors are arranged, with anequal inter-sensor spacing, on three-quoters of the surface of a spherical helmet
having a 13 cm radius.

19

(a)

(b)

Figure 3: Two sets of examples of source reconstruction results where the twosources are 3 cm apart. Maximum intensity projections of the three-dimensionalsource power maps onto the transverse, sagittal, and coronal planes are shown,respectively, in the left, middle, and right panels. The contours indicate therelative power of the reconstructed sources. (a)A relatively high SNR case
(SNR of 2) where the two sources are resolved. (b) A low SNR case (SNR of0.5) where the two sources are not resolved. The crosses show the assumed
locations of the two sources, and the blank circles in (b) show the location of adetected local peak.

20

Hit rate

1

0.5

0

False detection rate

1

0.5

0

A-prime

1

0.5

0
0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0.5

inter-source distance (cm)

(a)

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

inter-source distance (cm)(b)

Figure 4: Plots of the hit rate (top), false detection rate (middle), and Aprime metric (bottom) with respect to the inter-source distance. These plots
are obtained with the array of 160 sensors arranged on a spherical helmet with a
13 cm radius. (a) The source intensity is set to 60 nAm. (b) The source intensity
is set to 6 nAm. The horizontal broken line in the A-prime plot shows the levelof Ap = 0.75, and the vertical broken line shows the resolution determined as
the inter-source distance that gives Ap = 0.75. The error bars indicate the 95%
confidence interval.

21

False detection rate

1
Hit rate

0.8
0.6
0.4
0.2
0
−0.2
0.5

1

1.5

2 2.5 3 3.5 4 4.5
inter−source distance (cm)

5

5.5

6

1

1.5

2 2.5 3 3.5 4 4.5
inter−source distance (cm)

5

5.5

6

1
0.8
0.6
0.4
0.2
0
−0.2
0.5

1

1.5

2 2.5 3 3.5 4 4.5
inter−source distance (cm)

5

5.5

6

1
A−prime

0.8
0.6
0.4
0.2
0
−0.2
0.5

Figure 5: Plots of hit rate (top-left), false detection rate (top-right), and Aprime metric (bottom), with respect to the inter-source distance. These plots
are obtained using sLORETA source reconstruction algorithm. The array of 160
sensors is used and the source intensity is set to 400 nAm, which corresponds
to SNR of 20. The horizontal broken line in the A-prime plot shows the level of
Ap = 0.75. The error bars indicate the 95% confidence interval.

22

6

5

resolution (㎝)

4

3

2

1

0
40

source intensity (nAm)

20

SNR

10
5
2
1
0.5
0.2
4

6

8 10

20

40

60 80100200

400source intencity (nAm)

Figure 6: Resolution-source-intensity-SNR plot for the array of 160 sensors
arranged on a sperical helmet with a 13 cm radius. The upper panel shows theplot of resolution versus source intensity and the lower panel shows the plotof SNR versus source intensity. The error bars indicates the 95% confidence
interval. The broken line in the resolution plot indicates the regression results
obtained using Eq. (7) fitted to the plot for the range of source intensity between
10 and 100 nAm.

23

6

80 sensors
160 sensors
320 sensors
640 sensors

resolution (cm)

5
4
3
2
1
0

(a)

4

6

8 10

20

40

60

80 100200

400

standadized resolution

source intensity (nAm)(b)

80 sensors
160 sensors
320 sensors
640 sensors

1.2

1

0.8

0.6

4

6

8 10

20

40

60

80 100200400source intensity (nAm)

Figure 7: (a) Plots of resolution obtained using the four types of sensor arraysin Fig. 2. The error bars indicates the 95% confidence interval. (b) Plots ofthe modeled standardized resolution obtained using the regression analysis in
Eq. (8).

24

effective resolution

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5

10 2

10 3

number of sensors

Figure 8: Plot of the effective standardized resolution, β(k), with respect tothe number of sensors k. The broken line indicates the line fitted by using thesecond-order regression model in Eq. (10).

25

6

11.5cm13cm
10cm

resolution (㎝)

5
4
3
2
1
0

4

6

8 10

15 20

40

60 80100200

400source intensity (nAm)

Figure 9: Plots of the resolution for the sensor system with 160 sensors forhelmet sizes of 10, 11.5, and 13 cm. The plot for the 13 cm radius is the sameplot as in Fig. 6.

680 sensors
160 sensors
320 sensors
640 sensors

resolution (cm)580 sensors
160 sensors
320 sensors
640 sensors

4

3

2

1

modeled resolution

0

1

0.8

0.6

0.4

(a)

6

8 10

15 20

40

60 80100200

400source intensity (nAm)

6

(b)

8 10

20

40

60 80100200

400source intensity (nAm)

Figure 10: Plots of the resolution (top panel) and modeled standardized resolution xb(k) (I) (bottom panel) for arrays of 80, 160, 320, and 640 sensors. (a)Helmet radius of 11.5 cm. (b) Helmet radius of 10 cm.

26

13cm
11.5cm10cm

1.2

β(k):effective resolution

1.1
1
0.9
0.8
0.7
0.6
0.5
0.4

2

3

10

10

number of sensors

Figure 11: Plots of the effective standardized resolution β(k) with respect tothe number of sensors k for helmet sizes of 10, 11.5, and 13 cm. The plot for
the helmet size of 13 cm is the same as that in Fig. 8.

6resolution (cm)5

B=1.6cm
B=3.2cm
B=5cm
magnetometer

B=1.6cm
B=3.2cm
B=5cm
magnetometer

B=1.6cm
B=3.2cm
B=5cm
magnetometer

B=1.6cm
B=3.2cm
B=5cm
magnetometer

4
3
2
1
0
6

resolution (cm)

5
4
3
2
1
0

4

6 8 102040 60 80100

200

400 4

source intencity (nAm)

6 8 102040 60 80100

200

400

source intencity (nAm)

Figure 12: Resolution plot for the gradiometer baseline equal to 1.6, 3.2, 5,
and ∞ cm. (The gradiometer with the baseline of ∞ cm indicates the magnetometer.) Results for the arrays with 80 sensors, 160 sensors, 320 sensors, and640 sensors are respectively shown in the top-left, top-right, bottom-left, andbottom-right panels.

27

z
e
f
g

x

y

Figure 13: Schematic view of (e, f , g) directions. The unit vector e represents
the radial direction. The unit vectors f and g represent the two tangential
directions, namely, the longitudinal and latitudinal directions.

28

6
5

resolution (㎝)432

f gradiometerg gradiometerorthogonal gradiometer1

axial gradiometer(160)axial gradiometer(320)06

f gradiometerg gradiometerorthogonal gradiometer5

axial gradiometer(160)axial gradiometer(320)resolution (㎝)

4
3
2
1
0

4

6

8 10

15

20

40

60

80 100200400source intencity (nAm)

Figure 14: Plot of the resolution for the three-types of planar gradiometer arrays,
as well as for the axial gradiometer arrays with the same baselines. The top
panel shows results for gradiometer arrays with a 1.6 cm baseline and the bottom
panel shows results for those with a 3.2 cm baseline. The sensor locations forthe planar gradiometer arrays are the same as those of the axial gradiometer
array with 160 sensors. The plot for the 320 axial gradiometer array is shownby the broken line. The sensors are arranged on a spherical helmet with a 13 cm
radius.

29

65resolution (cm)4

3

2
radial sensors
f-sensors
g-sensors
vector sensors

1

0

4

6

8 10

15 20

40

60 80100200

400source intencity (nAm)

Figure 15: The plots of resolution versus source-intensity for the radial sensor
array, the g sensor array, the f sensor array, and the vector sensor array. We
assume that the sensor locations are the same as those for the 160 sensor arraywith a radius of 13 cm.

30

3radial sensors
f-sensors
g-sensors
vector sensors

standadized resolution

2.5
2
1.5
1
8 10

(a)

20

40

60

80 1003400

radial sensors
f-sensors
g-sensors
vector sensors

2.5
standadized resolution

200

source intensity (nAm)

2
1.5
1
8 10

(b)

20

40

60

80 100200400source intensity (nAm)

Figure 16: The plots of the modeled standardized resolution, xb(k) (I), for theradial sensor array, the g-sensor array, the f -sensor array, and the vector sensor
array. (a) Plots obtained under the assumption that the radial and the tangential sensors are aligned on the same spherical helmet with a 13 cm radius. (b)Plots obtained under the assumption that the radial sensors are aligned on aspherical helmet with a radius of 13 cm and tangential sensors on a sphericalhelmet with a radius of 14 cm.

31

7
Y
EL-G
EL-G+M

6

resolution (cm)

5
4
3
2
1

16 4
10

6

8

15 20

40

60 80100

200

6

8

15 20

40

60 80100

200

SNR

4
2
1
0.5

4

source intensity (nAm)
Figure 17: Plots of resolution and SNR versus source intensity for two existing
sensor systems. The plot labeled “Y” indicates the results for MEGvisionTM .
The plot labeled by “El-G” indicates the results for TRIUX when only theplanar gradiometer sensors are used. The plot labeled by “El-G+M” indicatesthe results for TRIUX when both gradiometer and magnetometer sensors areused. To compute SNR, we assume sensor noise with standard deviation of 50
fT.

32

Δ<2V
source 2

source 1
voxel
A

B

C

D

C

D

(a)

Δ=2V

A

B

Δ

(b)

Δ=2V

A

B

C

D

(c)

Δ>2V

V
A

B

C

D

(d)

(e)

Figure 18: Four scenarios with different ∆ and V settings are shown in (a)–(d)where ∆ is the inter-source distance and V is the size of voxels. (a) A case wherethe voxel size is greater than ∆/2. (b) A case where the voxel size is equal to∆/2 and the two sources can be detected. (c)A case where the voxel size isequal to ∆/2 but the two sources cannot be detected. (d) A case where thevoxel size is smaller than ∆/2. The locations of the two sources are indicatedby filled squares. Voxels are indicated by filled circles. The voxels neighboring
the sources are labeled “A”, “B”, “C”, and “D”. (e) Three-dimensional sourceand-voxel configuration corresponding to the one-dimensional case shown in (c).
The two sources are located at the centers of the cubic grids of voxels, and an
empty cube exists between the two cubes containing the sources.

33

voxelsourcevoxel

XYsource

V

(a)

(b)

Figure 19: (a) Three-dimensional cubic grid of voxels with a source locatedinside the cube. The source location is indicated by a small rectangle. The
distance between the source and one of the voxels is denoted X. (b) Twoneighboring cubic grids of voxels with a source located at the center of the lower
cube. The source location is indicated by a small rectangle. The voxel that isnearest the source but belongs to the upper cube is marked with a triangle, andthe distance between this voxel and the source is demoted Y34

(1,1)

1(F , H )Hit rate

H

(1,0)
0

F

1

False detection rate
Figure 20: The concept of A-prime in which the area under the ROC curve isapproximated by a rectangle whose corner’s coordinates are defined by (0, 0),(H, F ), (1, 1), and (1, 0).

35