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Extension of the visibility concept for EEG signal
processing
Valentin Debenay, Grégory Turbelin, Jean-Pierre Issartel, Philippe
Courmontagne, Amine Chellali, Marie-Hélène Ferrer
To cite this version:
Valentin Debenay, Grégory Turbelin, Jean-Pierre Issartel, Philippe Courmontagne, Amine Chellali,
et al.. Extension of the visibility concept for EEG signal processing. Journal of Neural Engineering,
2025, 22 (2), pp.026008. �10.1088/1741-2552/adb994�. �hal-04982685�
https://doi.org/10.1088/1741-2552/adb994
1
Extension of the visibility concept for EEG signal
processing
Valentin Debenay1,2,3,*, Grégory Turbelin1, Jean-Pierre Issartel1, Philippe
Courmontagne3, Amine Chellali2 and Marie-Hélène Ferrer2,3
1 Laboratoire de Mécanique et d’Énergétique d’Évry (LMEE, EA 3332), Université Évry Paris-Saclay,
40 rue du Pelvoux, 91020 Évry-Courcouronnes, France
2 Laboratoire Informatique, BioInformatique, Systèmes Complexes (IBISC, EA 4526), Université Évry
Paris-Saclay, 34 rue du Pelvoux, 91080 Évry-Courcouronnes, France
3 Institut de Recherche Biomédicale des Armées (IRBA), Service de Santé des Armées (SSA), 1 place
du général Valérie André, 91223 Brétigny-sur-Orge, France
* Author to whom any correspondence should be addressed.
E-mail: valentin.debenay@univ-evry.fr
Received 11 July 2024
Revised 7 January 2025
Accepted for publication 24 February 2025
Published 7 March 2025
Abstract
Objective. Visibility is an intrinsic property of any network of sensors that describes the
regions in which its measurement sensitivity is concentrated. Initially introduced to describe
the global spatial sensitivity of air pollution monitoring networks, we propose to extend the
concept of visibility to characterize the detection capabilities of electroencephalography
(EEG) systems utilized to measure brain electrical activity. Approach. In this paper, we
represent visibility within the brain as a field of symmetric 3×3 matrices, satisfying the so-
called “renormalization conditions” and interpreted as second-order tensors. A compact and
computationally efficient iterative algorithm is proposed for computing this tensor field. In
addition, we explain how to visualize and present the visibility information in an intuitive and
easily understandable way. Main results. The visibility concept is exploited to evaluate and
compare the ability of three consumer-grade EEG headsets to detect and localize an arbitrary
current distribution in the brain. Additionally, visibility is applied to derive an inverse
solution that can solve the Neuroelectromagnetic Inverse Problem (NIP) by reconstructing
focal brain sources from EEG data. Significance. Although the lead field function approach
can be employed to describe the sensitivity of individual electrodes from an EEG headset, this
paper extends the sensor network’s visibility concept to characterize the sensing capabilities
of a complete EEG system. The comparison between three consumer-grade EEG headsets
shows that the size of the low-visibility brain area decreases when the number of electrodes
used increases. In addition, we show that the source parameters are best estimated by the
inverse solution when they are oriented towards the maximum visibility direction.
Keywords: Electroencephalography, Neuroelectromagnetic Inverse Problem, Sensing capabilities, Source reconstruction,
Tensor field, Visibility, Visualization
2
1. Introduction
Sensor networks are systems consisting of interconnected
sensing nodes, each capable of measuring and communicating
data. These systems are employed in many fields: industrial
engineering, telecommunications, environmental monitoring,
threat detection, etc. [1]. Notably, they can be used to collect
neurophysiological data, such as the electrical activity of the
human brain. To do so, a network of electrodes is set on the
scalp to record voltage potentials resulting from current flow
in the brain. Such recordings are called
“electroencephalograms”. Electroencephalography (EEG) is a
noninvasive, relatively simple and inexpensive technique with
high temporal resolution (millisecond scale) [2] that has
proven helpful for the diagnosis and description of various
neurological disorders, such as epilepsy [3], amyotrophic
lateral sclerosis [4], multiple sclerosis [5] or for providing a
fast diagnosis of traumatic brain injury [6]. It is also
commonly used for EEG-based brain-computer interface
(BCI) applications [7], [8] and brain source reconstruction,
i.e., retrieving from measurements the neural origin, strength,
and orientation of electrical sources that generated them [9],
[10], [11].
EEG systems used for standard clinical recordings typically
involve a basic array of around twenty electrodes [12]. It is
recognized that increasing the number of electrodes provides
greater accuracy in source estimation [11]. Therefore, high-
density electrode arrays (from 64 to more than 300 electrodes)
are generally recommended to infer the location of the brain
area that generates the neuronal activity measured on the
scalp. In this case, the scalp electrodes are positioned
following globally recognized configurations, such as the
standard international 10-10 or 10-5 systems that enable,
respectively, the use of up to 74 and 345 electrode locations
[9].
Thanks to significant advances in microelectronic
biosensing technology, reliable EEG systems with fewer
sensors have recently emerged and are being used increasingly
in research [13], [14]. However, as decreasing the number of
electrodes reduces the covered area within the brain, the
amount of information that could be used for source term
estimation also diminishes. While it is possible to characterize
the sensitivity of pairs of electrodes by using the “half
sensitivity volume” (HSV) [15], [16], there is currently no
metric available to characterize the sensing capabilities of an
EEG system as a whole. Therefore, the question we aim to
answer in this paper is the following: how could we
objectively characterize the information a given sensor
network can provide? In particular, how could we evaluate the
sensing capabilities of an electrode system? The “visibility”
concept, proposed and justified by Issartel [17], can be used to
address this issue. Visibility can be defined as a collective
property of sensor networks. Based solely on the mathematical
concept of “adjoint functions” [18], [19], this scalar field was
initially employed to characterize the global spatial sensitivity
of air pollution monitoring networks. It has been successfully
used as a priori information to reconstruct emitting sources of
pollutants from measured concentration data [20], [21].
Since an EEG setup placed on the scalp can be viewed as a
sensor network having its own global sensitivity to the
location and the orientation of the brain’s current sources, it
can be characterized by its visibility. However, a direct
transposition of the existing theory is not possible: while in
atmospheric physics, the source producing concentration
measurements is represented by a scalar function [22], the
electrical source in the brain is represented by a vectorial
function [23], [24]. Consequently, the visibility of EEG
systems has to be characterized by a field of second-order
tensors rather than a zero-order (scalar) one. Because such a
mathematical object is quite difficult to visualize and interpret,
one of this work’s main challenges consists in presenting the
tensor field information in an intuitive and easily
understandable way.
The contributions of this proof of concept are as follows:
we present the new EEG visibility tensor field concept and
provide clear and intelligible interpretations of it. We also
propose an efficient algorithm to compute it quickly. In
addition, we explain how to visualize and present the visibility
information in an intuitive and easily understandable form.
Then, it is used to evaluate and compare the ability of various
EEG systems to detect and localize an arbitrary current
distribution in the brain. It is also used to derive an inverse
solution to reconstruct focal brain sources from EEG data.
The paper is organized as follows: Section 2 discusses the
theoretical background employed in this article. Section 3
provides an algorithm to compute the visibility tensors and
offers some recommendations on visualizing and presenting
it. Results are reported in Section 4 to illustrate the concept.
Section 5 discusses how they compare to prior efforts and their
future use.
2. Theoretical background
2.1 The scalar visibility
Suppose we have, on the one hand, a network of a finite
number of sensors measuring a phenomenon over a region of
interest and, on the other hand, the sources of this
phenomenon, which can be located anywhere within that
region. How well can we expect the network to accurately
sense the phenomenon when it originates from a specific
location? To answer this question, one should use, for each
point within the region, a metric to quantify the sensing
capabilities of the network. This subsection aims to introduce
and explain one such metric, which we refer to as “scalar
visibility”.
Throughout this paper, boldface Romans refer to matrices;
boldface italics indicate vectors in
( 3)
nnR
; lower-case
3
italics accented by a right arrow refer to vectors in
3
R
(e.g.
x
the position vector), and lower-case italics indicate scalars.
The concept of “scalar visibility” has been progressively
developed from 2001 onwards [17] within the atmospheric
physics framework to identify pollution sources, each
represented as a scalar function
()sx
, from a limited number
of concentration measurements
12
, , , m
collected by
sensors placed at positions
, 1,2,...,
i
r i m=
. The theory is
described here for a continuous emitting source but is easily
extendable to time-dependent sources.
Using a scalar product, a mutual correspondence between
the source emission function and the observations can be
defined in the space domain
:
( ) ( , )
ii
µ s x a r x dx
=
with i = 1, 2, …, m
(1)
In equation (1), each
( , ) ( )
ii
a r x a x=
is an “adjoint” (or
“influence”) function describing the individual sensitivity of
the ith sensor to all locations in the source space [25]. The m
equations (1) may be gathered in vector form as:
11
22
( ) ( ) ,
()
()
where and ( )
()
mm
s x x dx
µ a x
µ a x
x
µ a x
=
==
µa
µa
(2)
where
()xa
, the kernel of the first-kind Fredholm integral
equation (2), establishes a forward relationship from the
unknown source
()sx
to the limited set of measurements
μ
collected by the network of sensors.
In equation (2), all active sources in the domain are
summed considering their distance to the monitoring network.
Since the adjoint functions decrease rapidly with respect to
x
, unraveling the effects of distance and source strength is
known to be a challenging task, especially in the presence of
noise [26], [27]. As a matter of fact, any interpretation of the
measurements based on equation (2) will favor sources close
to the monitoring network over the ones located further away.
To mitigate this tendency, a scalar field, denoted
f( )x
, can be
applied in equation (2) as a weighting function that is large
when
()xa
is large to compensate for it. This is used to rewrite
the measurement with a new weighted scalar product in which
f( )x dx
is the new elementary volume of
:
ff
()
( ) ( )f( ) , where ( ) f( )
x
s x x x dx x x
==
a
µ a a
(3)
Naturally, the issue of selecting an appropriate function
f
arises. The choice of
f
is an assumption that affects the
interpretation of the measurements. It is well known that
entropy is a measure of uncertainty: higher entropy means
more uncertainty. Entropy is also associated with the average
amount of information conveyed by observations: the more
uncertain these observations are, the more informative they
are. Issartel [17] explained that the information unduly
introduced by
f
results in an excess of entropy that leads to
artifacts in the data interpretation. He showed that the
minimization of this excess can be framed as an optimization
problem with the entropic criterion
( )
( )
10 f
log det H
as the
cost function, where
f
H
is the Gram matrix defined as:
f( ) ( )
f( )
T
xx
dx
x
=
Haa
(4)
where the superscript “T” denotes the transposition. The
terms of
f
H
are:
f( ) ( )
f( )
ij
ij
a x a x
H dx
x
=
(5)
Over the infinite amount of possibilities, one function
f( )x
will ensure the minimization of the entropic criterion. This
function is known as the optimal weighting function and is
denoted by
. Issartel et al. [20] demonstrated that
satisfies
the following three properties called “renormalization
conditions”:
1
( ) ( ) 0
( ) ( ) ,
( ) ( ) ( ) 1,
T
ix
ii x dx m
iii x x x
−
=
=
Haa
(6)
where
H
is the matrix defined in equation (4).
From a physics perspective,
()x
is an intrinsic property of
any network of sensors (considered as a whole) that
characterizes the regions of space in which its measurement
sensitivity is concentrated. In the sensor coverage area, the
regions associated with high values of
are located close to
the sensors and/or monitored by multiple sensors
simultaneously. In contrast,
is low in the regions located far
away from the sensors and null outside the sensor coverage
area. As a consequence, in equation (3), for equally distributed
sources over
, the “well-seen” regions characterized by
high values of
would be “magnified” as opposed to the
more distant ones. In other words, the geometry weighted by
can be interpreted as the geometry apparent to the
4
monitoring network, making things appear “bigger” or
“smaller”, “sharp” or “flat”, depending on the distance. This
is why Issartel et al. [20] coined the term “visibility” in
connection with eyesight.
So far,
has been mainly used as a priori information to
build, from the measurements, a unique weighted minimum-
norm source estimate [22]:
1
ˆ( ) ( )
T
s x x
−
=Hµa
(7)
This so-called “renormalized inverse solution” is linearly
defined, and among all sources that fit the observed data, it is
the minimally surprising one. The entropic criterion used to
define
also deals with under-determinacy: it prioritizes the
regions in proportion to their visibility. As a result, the quality
of the estimate in poorly seen areas may seem low. This does
not matter, as signals originating from these regions have a
negligible probability of being sensed by the sensors. This is
the price for reliability in the regions that are well-seen.
However, since computing
is a goal in itself, one can
choose to compute it without necessarily searching for a
solution to an inverse problem. For instance, one can aim to
use it as a criterion for comparing various sensing systems
configurations or optimize the number of sensors and the
layout of a given monitoring network to maximize the total
“well seen” area.
In the following sections of this paper, we intend to apply
the visibility concept, on the one hand, to evaluate and
compare the sensing capabilities of several EEG systems, and
on the other hand, to solve the neuroelectromagnetic inverse
problem (NIP), i.e., to retrieve the electrical sources that
produce observed EEG signals. The first step, presented in the
next subsection, aims to extend the visibility theory to describe
the sensitivity of sensor networks when the spatial distribution
of the source is modeled as a vector field rather than a scalar
one. This task is not trivial since visibility must be
characterized by a field of second-order tensors instead of a
zero-order (scalar) one.
2.2 Extension of the visibility concept
While in atmospheric physics, the source that produces
concentration measurements is represented by a scalar
function, this section deals with the case of a source described
as a field of electric dipoles [23], [24], represented by a
vectorial function
()jx
defined through
with values in
3
. Without loss of generality, the study will be focused on EEG
data. In this particular case, the spatial domain corresponds
with the head, denoted
H
, and the source activity is modeled,
at each point, as a dipole (vector) with components
( ), ( ), ( )
x y z
j x j x j x
respectively along the sagittal, frontal,
and longitudinal axes of a Cartesian coordinate system which
origin is at the center of the brain.
EEG consists of recording, in time, the brain's electrical
activity using m electrodes placed on the scalp
H
, at
positions
i
r
,
1,2,...,im=
. To that, at a given sampling rate,
12
, , , m
µ µ µ
potential differences are performed between
these electrodes and another one, called "reference", placed at
a point
0
r
. The linear dependence of the measurements with
respect to the current source allows any potential difference to
be written, at each time step, as a scalar product:
0
( , , ) ( ) , with 1,2,...,
ii
µ a r r x j x dx i m= =
H
(8)
The
0 0 0 0 0
( , , ) ( , ) ( , ) ( , ) ( , ) T
x y z
i i i i i
a r r x a r x a r x a r x a r x
==
functions are the (vector) lead field functions of the m
electrodes. While they enable a linear mapping between the
electric sources within the brain and the scalp recordings, they
also describe the spatial sensitivity of each individual sensor.
Equation (8) implies that, for an arbitrary head model, the lead
field functions may be computed, point by point, by placing a
unit dipole at
x
oriented sequentially along each of the three
orthogonal directions and solving the forward problem in EEG
to obtain the m potential differences at sensor locations [28].
Another option consists in computing the lead field from a
sensor point of view, i.e., solving a so-called “adjoint
equation” for each sensor that directly gives the values of the
lead field for the whole domain [19]. Like any adjoint
function, they are functions of
i
r
,
0
r
and
x
, but they also
implicitly depend upon the physical properties of the various
head tissues. From now on, for readability reasons, the
dependency on the position of the reference electrode will be
omitted in the notation.
Once the lead field functions are obtained, the m equations
(8) may be gathered in matrix-vector form as:
11
1
1
222
2
( ) ( ) ,
( ) ( ) ( )
( ) ( )
()
where and ( )
( ) ( )
()
y
xz
x y z
xz
y
mmm
m
x j x dx
a x a x a x
µ
µa x a x
ax
x
µa x a x
ax
=
==
a
a
µ
µ
H
(9)
where
()xa
, the kernel of the first-kind Fredholm integral
equation (9), establishes a forward relationship between the
current sources in the brain
()jx
and a set of potential
differences
µ
measured by surface electrodes placed on the
scalp. In this context, visibility is obtained through a “three-
dimensional renormalization” process. It amounts to
optimizing a function
()xf
with values in the set of
33
symmetric positive matrices. This is used as a weighting
5
function to rewrite the measurement equation with a new
weighted scalar product and correspondingly modified lead
field functions:
1
( ) ( ) ( ) ,
where ( ) ( ) ( )
x x j x dx
x x x
−
=
=
f
f
af
a a f
µH
(10)
The weight function
()xf
is associated with a
mm
symmetric positive definite Gram matrix:
1
( ) ( ) ( ) ,
i.e. ( ) ( ) ( )
ij
T
f i j
x x x dx
H x a x a x dx
−
=
=
f f f
H a f a
f
H
H
(11)
The optimal weight function
()xφ
can be obtained from a
minimum entropy principle [29] by solving the constrained
optimization problem:
( )
()
( ) argmindet( ),
subject to Tr ( )
x
x
x dx m
=
=
f
f
φH
f
H
(12)
where det denotes the determinant, and Tr is the trace.
()xφ
is equivalently determined by the renormalization constraints:
( )
13
( ) Tr ( )
( ) , ( ) ( )
T
i x dx m
ii x x x
−
=
=
φ φ φ
φ
a H a I
H
H
(13)
The renormalized estimate of the source activity at location
x
, is then defined as:
11
ˆ( ) ( ) ( )
T
j x x x
−−
=φ
φ a H µ
(14)
Visibility is not affected by re-referencing (i.e., it is
independent of the choice of the reference upon the m+1
electrodes). However, it is dependent on the domain's physical
properties (through the lead field functions) and the recording
system structure (number and location of the sensors). With
respect to any chosen coordinate system,
()xφ
is represented
as a symmetric matrix of
33
real numbers (hence, we only
need six instead of nine components to describe it). In
equation (10), the weight function takes the vectors
()
i
ax
as
input and gives weighted vectors as output. Thus, visibility
can be interpreted as a second-order tensor field describing
how well the sensor system monitors the various locations and
directions inside the brain.
()xφ
can also be described by its eigenvalues
1 3 3
( ( ), ( ), ( ))xxx
such that
1 2 3 0
, and their
corresponding eigenvectors
1 2 3
( ( ), ( ), ( ))u x u x u x
. At each
position, eigenvalues represent the magnitude of visibility,
and the corresponding eigenvectors reflect the directions of
maximal and minimal visibility. Eigenvalues can be used to
process scalar indices of visibility, the most convenient one
being their average. It corresponds with the mean visibility
()x
and provides information regarding the global visibility
of the electrode system, at position
x
, regardless of the
direction:
( )
1 2 3
( ) ( ) ( ) 1
( ) Tr ( )
33
x x x
xx
++
==φ
(15)
2.3 Physical interpretations for EEG signal analysis
While the scalar visibility field defined in Subsection 2.1 is
a map assigning to every point in the scalar source space the
scalar (zero-order tensor)
()x
, the visibility tensor field over
the brain is a map that assigns to every point in the vectorial
source space the second-order tensor
()xφ
. Similarly to
,
φ
can be interpreted both in terms of network sensitivity
(sensing capabilities) and source identifiability.
Classically, we can define the sensitivity of a single sensor
through the notion of lead field function. Here, the visibility
tensor concept is used to determine the sensitivity of a
monitoring network as a whole. Indeed, the visibility tensor
field conveys information about the sensitivity of an EEG
headset (which can be seen as a scalp-attached network of
electrodes) and can be used to define its capability to sense the
source space. The mean visibility, computed in equation (15)
as a function of position in the brain volume, characterizes the
zones of the source space sensed by the network of electrodes.
Thus, the mean visibility map can address two fundamental
questions: i) Which regions of the source space are effectively
covered by the EEG system? ii) How well is the system able
to sense these various regions? Yet, the orientation of the
sensed sources, in addition to their location, is a critical factor
in determining the sensitivity of an EEG system. Therefore,
by computing the eigenvalues of the visibility tensors, one can
determine the source orientation the EEG sensor array is the
most sensitive to (directions of the largest eigenvalues). This
can answer the following question: What are the source’s
orientations that are preferentially sensed or that will be
observed with an enhanced probability?
Unlike sensitivity, which is a property of the sensor
network, we define identifiability as a property of the source:
it indicates how correctly inverse methods can characterize the
electrical source at every solution point. Since visibility
indicates how well a current source can be observed by a given
electrode configuration (one ordinary meaning of “visibility”
is “ability to be seen”), it also determines how precisely the
source parameters (location, intensity, and orientation) can be
6
reconstructed based on knowledge of the set of measurements,
e.g., using equation (14). Uncertainties in the reconstruction
depend on both the position and the orientation of the source,
relative to the measuring system [26]. Consequently, they are
functions of visibility. Therefore, mean sensitivity maps could
quantify the strong depth dependency of uncertainties. At the
same time, the direction of the largest eigenvalue of the
visibility tensors could highlight the direction in which
sources must be oriented to be maximally detected and better
identified.
3. Computational implementation and visualization
3.1 Computation of the Lead field functions
We performed the computation of the lead fields functions
using the Finite Element Method (FEM). To do so, we used
realistic head models. These 3D models were created by
segmenting high-resolution Magnetic Resonance Images
(MRI) using the Brainstorm [30] toolkit
(http://neuroimage.usc.edu/brainstorm) on ©MATLAB. The
MRI files we acquired with a 1.5-Tesla scanner by the
McConnell Brain Imaging Center (McGill University,
Canada). Then, we meshed the 3D head model with
tetrahedral elements using the iso2mesh toolbox developed by
Fang and Boas [31]. Each of the 65113 tetrahedrons of our
volume mesh is associated with one of the three types of
biological tissue we considered: brain, skull, and scalp. We
assigned widely used default isotropic conductivity values to
the three tissues: 0.33 S/m for the brain, 0.43 S/m for the scalp,
and 0.008 S/m for the skull (40 times smaller than for the
brain). We opted for this simple 3-layer model for this proof-
of-concept phase as it enables a functional trade-off between
simulation accuracy and computational cost. Finally, for a
given electrode configuration, the FEM calculation of the lead
field functions is performed by using the Brainstorm-
DUNEuro pipeline [32], accessible through a user-friendly
GUI. Then, Brainstorm interpolates the results on a grid of
points
k
x
,
1kn=
that samples the brain volume. The
3mn
lead field matrix is obtained as
12 n
=
A a a a
with
( ), 1
kk
x k n==aa
.
3.2 Computation of the visibility tensors
To iteratively compute
φ
, the integral in equation (9) is
approximated by a Riemann sum:
1
( ) ( )
n
k k k
k
x x j x
=
=
aµ
(16)
where
k
x
is the representative elementary volume around
k
x
. Let
3n
Rj
be the vector
12
( ) ( ) ( ) T
n
j x j x j x
and
A
the matrix
1 1 2 2 nn
x x x
a a a
. Then, in matrix
notation, equation (9) becomes:
=Aμj
(17)
Similarly, in matrix notation,
φ
H
(
mm
in size) can be
expressed as:
T
=
φ
HAΦA
(18)
where
Φ
is a
33nn
symmetric block-diagonal matrix
named “visibility matrix” and defined as:
1
33
00
0 0 ,
00
where ( ) , with 1,...,
n
kk
x k n
= =
φ
Φ=
φ
φφ
(19)
The visibility tensors are the n blocks of
Φ
. They satisfy
the renormalization conditions (discrete counterpart of
equation (13)):
1
1 1 1 3
( ) Tr
( ) , with 1,...,
n
kk
k
T
k k k k
i x m
ii k n
=
− − −
=
==
φ
φ
φ a H a φ I
(20)
The resolution matrix
1 1 3 3T n n− −
=
φ
RΦ A H A R
is
introduced to consider condition (20). This matrix is
composed of
2
n
blocks of dimension
33
. When
Φ
satisfies
the renormalization requirements, then the
n
diagonal blocks
of the matrix
1−
RΦ
are equal to
3
I
. An algorithm based on
this property has been developed. It is derived from the one
proposed in atmospheric sciences to compute the scalar (or
1D) visibility [22]. The flowchart in Figure 1 graphically
represents it. It consists of two parts. The first one is the
computation of
1−
φ
H
, the second one is the computation of the
n diagonal blocks of
1−
RΦ
.
To obtain the optimal blocks
k
φ
of
Φ
, the following
iterative scheme is used:
( ) ( ) ( )
( )
( )
11
1
1T
k k k k k k
l l l l
l
−−
−
+=φ
φ φ φ a H a φ
(21)
where the subscript « l » denotes the iteration step. Noticing
that the term under the square root in (21) corresponds with
the kth diagonal block of
1−
RΦ
. Numerical tests using random
matrices show that the algorithm always converges uniformly
to the visibility tensors satisfying the renormalization
7
conditions and that convergence is observed from any initial
k
φ
provided no
0
)(k
φ
is singular for
1kn=
.
Figure 1. Flowchart describing the proposed algorithm.
The algorithm described in Figure 1 has been implemented
in ©MATLAB code. For a grid of 40290 points, it provides
results with an error of O(10-7) within about 25 iterations and
with a CPU time of 35 s (the computations were performed on
a desktop Intel Core I9-9900 CPU 3.10GHz CPU and 64GB
RAM).
3.3 Visualization of the visibility tensors
In physics and engineering, visualizing and presenting
tensor information in an intuitive and easily understandable
way is known to be a challenging task [33]. Prior to any
measurement, plotting 3D mean visibility maps is the most
straightforward visualization technique that can be used to
answer the following question: for a given sensor
configuration, which current source distribution can be
observed with an enhanced probability? Indeed, even if the
sources are equally likely from all parts of the brain, those in
regions characterized by high values of
()
k
x
would be
sensed with an enhanced probability as compared to the more
distant, deeper ones (concept of “detection potentiality”). It is
important to note that this probability should not be mistaken
for the one of a source occurring in these regions. By
extension, mean visibility also describes how well any source
can be localized and contains information about the resolution
of the recording system. Usually,
()
k
x
is the highest in the
regions close to the sensors, where the sensitivity functions of
the various electrodes have large values and/or are
overlapping. Then, it rapidly decreases as a function of the
distance from the scalp electrodes. Since mean visibility
undergoes relatively significant changes over the conductive
domain, it is convenient to express its logarithm in decibels:
10
20log ( )
dB
=
(22)
For visualization, we picked a highly contrasted color
palette so that anyone could instantly differentiate the low-
visibility (dark) areas from the high-visibility (bright) ones.
For illustrative purposes, Figure 2 shows the mean visibility
map for the consumer-grade ©ANT Neuro Waveguard
Original 32 headset. Its 32 electrodes are placed according to
the international 10-10 system so that it covers most of the
head surface. As illustrated, the anterior prefrontal cortex, as
well as areas located over the parietal and the occipital lobes,
belong to the brain regions best seen by the electrode network.
Meanwhile, the most rostral portions of both the superior and
the middle temporal gyrus appear as the most poorly seen area.
Figure 2. Mean visibility on the cortex surface for the ©ANT Neuro
Waveguard Original 32 headset, with its 32 electrodes colored in khaki. The
magnitude of the mean visibility, expressed in decibels (dB), is colored
according to the scale on the right. As can be seen, the greatest mean visibility
values (
80dB
) are reached in the anterior prefrontal cortex and various
areas distributed over the parietal and the occipital lobes. The lowest values (
60dB
) are observed in the most rostral portions of the superior and the
middle temporal gyrus.
However, this visualization technique has limitations
regarding the amount of information presented: it fails to fully
account for the fact that visibility is a field of tensors that
carries different values along different directions. This
drawback can be overcome by displaying the tensors as
ellipsoids in a 3D grid. To do so, for each point of the discrete
domain, we: i) solve the eigenvalue problem; ii) create
ellipsoids with semiaxis lengths proportional to the square
root of the obtained eigenvalues; iii) align the ellipsoids along
with the direction of the eigenvectors associated with the
eigenvalues.
8
Figure 3. (Left) Visibility 3D tensors, represented as ellipsoids, for the same
©ANT Neuro Waveguard Original 32 headset used in Figure 2. (Right) From
top to bottom: Frontal (at x=0.002m), sagittal (at y=0m), and transverse (at
z=0.06m) slices of the head model are displayed. The size and color of the
ellipsoid associated with each point of the discrete domain correspond to the
magnitude of the visibility. The directions of the ellipsoids’ major axis
indicate the directions of the maximum visibility (directions to which the EEG
system is sensitive the most). The color bar used here is similar to the one
applied in Figure 2. The closer to the electrodes the points of the discrete
domain are, the larger and brighter the ellipsoids associated with these points
are. In the right subfigure, some of the cortical ellipsoids are clearly oriented
toward the electrodes (the electrode positions are showcased in Figure 2).
In Figure 3, the computed ellipsoids' sizes, colors, shapes
and orientations allow us to visualize all the visibility
information available in the discrete domain for the consumer-
grade ©ANT Neuro Waveguard Original 32 headset. First, the
size and the color reflect the visibility level of the area in
which it is located. Indeed, the mean visibility is proportional
to the trace (see equation (15)), which is also, in tensor theory,
a measure of the tensor’s size. We can see that, while the
tensor field is mainly constituted of tiny spheres in the deeper
brain regions, large ellipsoids can be observed in the more
superficial areas. Second, the shape and orientation indicate
the direction the EEG system is sensitive to the most, which
also corresponds to the direction the source has to be oriented
along to be maximally detected (direction of maximum
visibility). In low visibility regions, we can see that the
ellipsoids are spheres (no preferential direction). In the high
visibility regions, where
1 2 3
( ) ( ) ( )
k k k
x x x
, they
appear as elongated ellipsoids oriented along their main axis,
which correspond with the directions of maximal visibility.
The main drawback to this method is that it requires solving
many eigenvalue problems. This computation may become
costly when the datasets are big (fine grids). Another one is
that the 3D maps quickly become unreadable since ellipsoids
tend to overlap. In this case, only a selected sample of data
may be plotted. For example, less than 5% of the computed
ellipsoids have been plotted on the left subfigure of Figure 3,
and 50% on the right subfigures, to avoid overplotting issues.
4. Results
4.1 Comparison of sensing capabilities
In this part, visibility is employed to evaluate and compare
the respective sensitivity of three consumer-grade EEG
devices. The first one is an ©Emotiv EPOC+ headset
(hereafter referred to as “EPOC14”). It possesses 14
electrodes, positioned according to the 10-20 international
system [34], and two CMS/DRL reference electrodes in
P3/P4. The second and third ones are Waveguard Original
headsets (hereafter referred to as “WAVE32” and
“WAVE64”). They have 32 and 64 electrodes, respectively,
positioned according to the 10-10 international system [35].
Two additional electrodes in AFz and CPz are used as the
ground and reference. In this study, we suppose the three EEG
systems possess equally accurate electrodes with similar
manufacturing quality.
The three electrode configurations are shown in Figure 4,
along with the corresponding cortical mean visibility maps as
well as coronal, sagittal and horizontal sectional views. It is
crucial to keep in mind the equation (22) where visibility is
expressed in decibels (dB), a logarithmic unit. Consequently,
a 6dB increase means doubling the visibility level, while a
20dB increase means multiplying it by 10.
Figure 4. Comparison of the surface and volume mean visibility maps, both
expressed in decibels (dB), over three consumer-grade EEG devices. From
top to bottom and from left to right: Frontal (at x=0.002m), sagittal (at y=0m),
and transverse (at z=0.06m) slices of the volume mean visibility for the
EPOC14, WAVE32 and WAVE64 headsets. Brighter colors indicate higher
magnitudes of mean visibility values. Increasing the number of electrodes
from 14 to 64 drastically enhances the maximum mean visibility value from
91dB with the EPOC14 to 106dB with the WAVE64.
9
Figure 4 shows the values of
()
k
x
displayed on the
cortical surface for the three EEG headsets. First, it can be
seen that the brain is better monitored overall by both ©ANT
Neuro Waveguard Original devices, with a relatively uniform
visibility distribution. Indeed, these devices’ electrodes offer
a much more uniform sampling of the scalp surface than the
©Emotiv EPOC+, whose electrodes are mainly positioned on
the frontal and occipital lobes. This confirms that visibility
depends on the placement of the electrodes, regardless of their
accuracy, and that using a large number of electrodes
positioned following internationally recognized
configurations (e.g., 10-20, 10-10, 10-5) provides adequate
visibility over all the brain regions. The maximum values are
located on the frontal and occipital lobes. This can be
explained by the head model we have selected: those are the
regions where the thickness of the scalp and skull layers are
the lowest. Second, it can be seen that visibility improves
when the number of electrodes increases: when the number of
electrodes rises from 14 to 64, the maximum value rises from
91dB to 106dB (which means a 5.6-fold increase). Besides,
the median mean visibility values are 77dB, 82dB and 88dB
for 14, 32 and 64 electrodes, respectively. In the future, more
work needs to be undertaken to assess whether this trend
remains when adding more electrodes by testing 128-electrode
and 256-electrode EEG devices.
Figure 5 shows the evolution of the mean visibility along
the intersections of the coronal, sagittal, and horizontal planes
for the three consumer-grade EEG devices. This figure shows
the rapid decrease (up to a factor of 0.1) of the mean visibility
as a function of the distance from the scalp. This confirms the
natural intuition that superficial cortical sources will be sensed
with a higher probability than distant, deeper ones. This also
suggests that detecting deep EEG sources is a difficult task
due to low visibility. Therefore, the detection probability of a
signal originating from deep regions is extremely low, even
when using 64 scalp electrodes. Consequently, brain activity
measured by these EEG systems should be assumed to be
generated in the shallower regions of the cerebral cortex.
Another way of highlighting this difficulty consists in setting
a threshold (the horizontal straight line in Figure 5) that
distinguishes well-seen (high-visibility) from poorly seen
(low-visibility) regions. Because visibility is a novel concept,
there is no actual reference to an objective threshold value in
the literature. Since 32 electrodes represent a reasonable trade-
off between the low-intensity EEG EPOC14 headset and the
high-density EEG WAVE64 headset [36], we have selected
the median mean visibility value for the WAVE32 headset
(82dB) in this study. With the EPOC14, around 80% of the
brain regions reach a mean visibility value below this
threshold, for only 20% of them with the WAVE64 headset.
Figure 5. Comparison of the evolution of the mean visibility, expressed in
decibels (dB), along the intersections of the frontal, sagittal, and transverse
planes denoted by the Ox, Oy, and Oz axes, between three consumer-grade
EEG devices. The curve colors are the ones used to plot the electrodes of the
three EEG devices in Figure 3. The threshold settled at 82dB separates the
well-seen regions of the brain from the poorly seen ones. The left and right
extremities of the curves along the x and y axes, as well as the right extremity
along the z axis, correspond to cortical regions. The middle parts of the curves
are associated with deep brain regions.
4.2 Solving the Neuroelectromagnetic Inverse Problem
This section uses the visibility field to implement the
inverse solution described by equation (14). Here, the goal is
not to perform a complete assessment of the inverse method,
but rather to report some results that illustrate its capabilities.
The main feature of this solution is that, in the case of
noiseless data originating from a focal source of any direction
(i.e.,
00
( ) ( )j x j x x
=−
where
(.)
is the Dirac delta
function), it allows an unbiased reconstruction of the source,
i.e., it gives an unbiased estimation of localization, intensity
and orientation of the dipole. Indeed, it can be shown (see
appendix) that:
1
0 0 0 0
ˆˆ
argmax ( ) and ( ) ( )
x
x j x j x j x
−
==φ
(23)
Since the ultimate proof for any inverse algorithm is its
practical validation the proposed solution has been tested and
evaluated using pseudo-EEG generated by a dipolar neural
source according to the following scenario. At various
positions in the source space, a dipole moment with
exponential decay and damped oscillations was simulated (the
dipole orientation and location were kept constant during the
time course):
( )
( 2 )
0
( , ) e 1 cos(2 )
kt
j x k t a fk t
−
= +
(24)
10
with
0,...,511k=
,
2t ms
,
0.5a=
,
2Hz
=
and
10Hzf=
. We selected this function as the “damped
Gaussian function” used by Lei and Liao [37] “looks like an
evoked potential”. Then, a “clean” EEG signal was computed
by multiplying the dipole moment time course by the
corresponding lead field matrix. As 64-electrode systems are
widely applied in experimental studies, we focused our
analysis on the WAVE64 headset. The clean signals of the
channels were corrupted with background noise. Following
Jonmohamadi et al. [38], the background EEG was obtained
from a subject in a resting state mode. The parameter a in
equation (24) has been adjusted to obtain a global signal-to-
noise ratio (SNR) varying in time from 30 to -4dB (here,
global SNR was defined as the mean ratio of the amplitude of
the signals to the amplitude of the background noise,
transformed in decibel scale). Note that the potential curves
obtained at each electrode follow the dipole moment time
course and can be roughly separated into two parts: In the first
one (first 0.4 seconds), the curves rapidly decrease, large
oscillations are visible, and the clean signal amplitude is
globally larger than the noise amplitude. In the second one
(from 0.4s to 1s), the amplitude is very low, near zero, and
mainly dominated by background noise.
The inversion procedure was performed fast and
satisfactorily for each dipole position and time step.
Additionally, three error measures were applied to determine
the performance of the inverse solution: Localization Error
(LE), i.e., distance of the estimated dipole location to that of
the actual location of the source (in m), Orientation Error
(OE), i.e., angle between the estimated orientation and that of
the true dipole (in degree) and Relative Intensity Error (RIE),
i.e., intensity difference divided by the actual intensity (in %).
As an example, results obtained with a dipole located in the
cerebral cortex and oriented towards the maximum visibility
direction are shown in Figure 6. We can see that the quality of
the estimation of the source parameters strongly depends on
the strength of the signal relative to background noise (relative
noise level). Indeed, in the first part of the time course, when
the signal amplitude is larger than the noise one, we can see
that excellent estimates for all the source parameters are
obtained. In the second part, as the background noise becomes
dominant, the inverse solution comes to be of unacceptable
quality. This is because all the simultaneously active spurious
sources from which the background noise originates affect the
source estimation process.
Figure 6. Dipole moment time course (top) and error measures when the
source is in the cerebral cortex and oriented towards the maximum visibility
direction. After 0.6s, the background noise amplitude is much larger than that
of the signal, and the inverse solution is of unacceptable quality.
Table 1 shows global results (mean values over the first part
of the time course) when the source is in high (occipital lobe),
medium, and low visibility (subcortical structures) regions.
We can see that excellent estimates for all the source
parameters are obtained when the source is in a well-seen area.
As expected, the errors increase as the source is displaced into
less visible regions, reaching their maximum in deep brain
areas. Moreover, the results confirm that the source
parameters are best estimated when they are better detected,
i.e., when they are oriented towards the maximum visibility
direction.
Table 1. Error measures were computed over the first 0.4s of the time course,
with the dipole oriented towards the maximum visibility direction (left) vs.
towards the random direction (right). The characteristic grid size is 0.005m.
Visibility
Maximum visibility direction
Random direction
LE [m]
RIE [%]
OE [°]
LE [m]
RIE [%]
OE [°]
High
0.003
6,27
3,92
0.005
7.32
7.45
Medium
0.008
14.87
14.07
0.018
29.09
16.26
Low
0.033
41.6
21.9
0.042
52.02
32.02
5. Discussion
The previous results demonstrate that the concept of
visibility tensor can be applied to describe the ability of any
given EEG system to observe the source space despite its
spatially blurred view and limited resolution. By extension, it
can also be used to describe how well source parameters can
be identified. This approach represents a novel way to use the
lead field functions to study both the network's sensitivity and
the sources’ identifiability. Previously, the lead field functions
were used to compute a metric known as “half sensitivity
volume” (HSV) [15], [16]. Yet, HSV only characterizes the
sensing capability of pairs of electrodes (including one
11
reference electrode): it corresponds to the volume of the
source region within the volume conductor where the
magnitude of a sensor's sensitivity is more than one-half of its
maximum value [39]. Therefore, the smaller the HSV, the
more the electrode sensitivity is concentrated. Based on the
same lead field functions, the visibility concept characterizes
the sensing capabilities of an EEG system as a whole. Thus,
unlike HSV, it can be considered a collective measure that
entirely and uniquely describes any network of electrodes.
This allows for the comparison of EEG system configurations
and could be helpful for determining an optimal electrode
arrangement to maximize the well-seen/minimize the poorly
seen area (e.g., how each of the 14 electrodes of the EPOC14
headset should be re-arranged to minimize the black area in
Figure 4). Following the work presented by Turbelin et al.
[21], mean visibility, associated with an efficient
combinatorial optimization algorithm, could be used to define
a cost function for finding the best positions on the scalp to
place the electrodes. Even if this last problem is
computationally challenging to solve, several good
combinatorial optimization algorithms to tackle it now exist
[40].
Next, as explained in Subsection 4.2, visibility can be used
to build an inverse solution to the NIP. Since this solution can
be computed quickly and efficiently, it is expected to be useful
for real-time brain monitoring, neurofeedback, or brain-
computer interface (BCI) applications. EEG-based BCIs are
communication systems that translate neuroelectric patterns
into commands for interactive applications without requiring
any peripheral muscular activity [7], [8]. It has been
demonstrated that applying EEG source imaging (ESI)
techniques can facilitate the classification of motor imagery
(MI) tasks by converting the smeared scalp potentials into
source distributions within the brain [41], [42]. However, a
theoretical comparison with other well-known ESI methods is
required to understand its efficiency and properties fully.
Finally, within the estimation theory framework, one can
make a parallel between visibility and well-known
parameters, such as variance, bias, mean square error (MSE),
and spread of the estimated values, that characterize the
uncertainty one has in estimations. Indeed, a large visibility
value can be associated with low variance, low bias, and low
mean square error. The spread is proportional to the inverse of
the visibility. Nevertheless, the computation of variance, bias,
and MSE requires a priori knowledge about the ground truth,
which is not the case with visibility. In addition, to be
computed, these statistical parameters generally require a high
number of observations, multiple simulations, or both.
However, in the case of visibility, only one iteration is needed
to get the same amount of information. As visibility provides
new insights into understanding uncertainty, it could provide
a degree of confidence in inverse problems, as well as in BCI
systems’ interpretations of brain activity patterns.
These topics will be discussed in further detail in future
work.
6. Conclusion
In this paper, we have introduced the concept of visibility
tensor. The visibility tensor field is obtained from the lead
field functions that describe the sensitivity matrix of each
individual electrode. Visibility involves a large symmetric
block-diagonal matrix
Φ
whose blocks (tensors) satisfy the
so-called “renormalization conditions”. A compact and
computationally efficient iterative algorithm for computing
this matrix has been proposed, and solutions to visualize and
present the visibility information in an intuitive and easily
understandable form have been given. Tensors have been
interpreted in terms of sensitivity, which describes the
capability of a network of electrodes to measure data in the
source space, and identifiability, which indicates if inverse
algorithms can characterize the electrical source at every
solution point despite the spatial blurring. Here, the visibility
concept has been used: i) to evaluate and compare the ability
of three EEG systems to detect and localize any current
distributions in the brain (see Figures 2, 3, 4 and 5); ii) to build
a solution to the source reconstruction problem (see Figure 6).
In future works, the visibility comparison conducted
between the three EEG systems should be extended to a larger
number of headsets, in particular 128-electrode and 256-
electrode EEG devices, in order to determine whether the
trend observed in this paper is confirmed. Also, it could be
valuable to work on the generalization of the visibility concept
to other scientific fields that process experimental data.
Appendix
In the case of a focal source localized at
0
x
,
00
( ) ( )j x j x x
=−
, equation (9) gives:
00
()xj=aµ
(25)
Consequently, the renormalized estimate of the source
activity at any location in the brain (equation (14)), may be
written as:
11
00
ˆ( ) ( ) ( ) ( )
T
j x x x x j
−−
=φ
φ a H a
(26)
So, one has:
21 2 1
0 0 0 0 0
ˆ( ) ( ) ( ) ( ) ( ) ( )
T T T
j x j x x x x x j for x x
− − −
=
φφ
a H a φ a H a
(27)
and, by making use of equation (13):
12
22
0 0 0 0
ˆ( ) ( )
T
j x j x j=φ
(28)
By using Cauchy-Schwarz inequality, one has:
21 2 1 2 1
0 0 0 0
1
0 0 0 0
21 2 2 2 1
0 0 0 0
2
0 0 0
2
0
ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
()
ˆ()
T T T T
TT
T T T
T
j x j x x x x x x x x j
j x x j
j x j x x x x r x x j
j x j
j x j
− − − − −
−
− − − −
φφφ
φ
φφ
a H a φ a H a φ a H a
a H a
a H a φ φ φ a H a
φ
1 2 1
0 0 0 0
2
0
ˆ
( ) ( ) ( ) ( ) ( ) ( )
ˆ ˆ ˆ
( ) ( ) ( )
T T T
x x r x x j j x
j x j x j x
− − −
φφ
a H a φ a H a
(29)
Therefore
00
ˆˆ
( ) ( ) j x j x x x
, i.e., the renormalized
estimate has its maximum norm exactly at the location of the
focal source:
0ˆ
argmax ( )
x
x j x=
(30)
and from (26):
1
0 0 0
ˆ
( ) ( )j x j x
−
=φ
(31)
This means that, in the absence of noise,
100
ˆ
( ) ( )x j x
−
φ
is
an unbiased estimator of the dipole at the estimated source
location.
Authors’ contribution
Valentin Debenay: Software, Validation, Investigation,
Writing – original draft, review & editing. Grégory Turbelin:
Conceptualization, Methodology, Visualization, Supervision,
Writing - original draft, review & editing. Jean Pierre Issartel:
Conceptualization, Formal analysis, Writing – review &
editing. Philippe Courmontagne: Methodology, Writing –
review & editing. Amine Chellali: Supervision, Writing –
review & editing, Project administration. Marie Hélène Ferrer:
Supervision, Writing – review & editing, resources.
Acknowledgments
The authors would like to express their gratitude towards
the Editor and the Reviewers for their time and valuable
comments that helped improve this paper.
The authors also wish to thank the following researchers:
Xavier Busch (French Defence Procurement Agency, DGA
Maîtrise NRBC), who helped us to develop a better
understanding of the 3D visibility concept, and Sébastien
Grosjean (former postdoc fellow at LMEE) for their helpful
collaboration to the project.
This work was supported by the LabEx LaSIPS (ANR-10-
LABX-0032-LaSIPS) managed by the French National
Research Agency under the “Investissements d'avenir”
program (ANR-11-IDEX-0003-02), and by the Paris Île-de-
France Region (DIM RFSI, grant 20002717).
Conflict of interest
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
ORCID iDs
Valentin Debenay: https://orcid.org/0000-0002-3852-0488
Grégory Turbelin: https://orcid.org/0000-0001-9929-3090
Jean-Pierre Issartel: https://orcid.org/0000-0002-1929-8535
Philippe Courmontagne: https://orcid.org/0009-0002-6858-
7779
Amine Chellali: https://orcid.org/0000-0002-6143-5898
Marie-Hélène Ferrer: https://orcid.org/0000-0003-2132-9588
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