1430 ieee transactions on biomedical engineering, vol. …ldupre/2011_2.pdf · 1430 ieee...

1430 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 58, NO. 5, MAY 2011

Reduced Conductivity Dependence Method forIncrease of Dipole Localization Accuracy

in the EEG Inverse ProblemBertrand Russel Yitembe, Guillaume Crevecoeur*, Roger Van Keer, and Luc Dupre

Abstract—The EEG is a neurological diagnostic tool with hightemporal resolution. However, when solving the EEG inverse prob-lem, its localization accuracy is limited because of noise in mea-surements and available uncertainties of the conductivity value inthe forward model evaluations. This paper proposes the reducedconductivity dependence (RCD) method for decreasing the local-ization error in EEG source analysis by limiting the propagationof the uncertain conductivity values to the solutions of the inverseproblem. We redefine the traditional EEG cost function, and incontrast to previous approaches, we introduce a selection proce-dure of the EEG potentials. The selected potentials are, as low aspossible, affected by the uncertainties of the conductivity whensolving the inverse problem. We validate the methodology on thewidely used three-shell spherical head model with a single electricaldipole and multiple dipoles as source model. The proposed RCDmethod enhances the source localization accuracy with a factorranging between 2 and 4, dependent on the dipole location and thenoise in measurements.

Index Terms—Conductivity, EEG source analysis, inverse prob-lems, source localization, uncertainty.

I. INTRODUCTION

THE EEG is a medical imaging technique for the diag-nosis of neurological disorders. By placing metal elec-

trodes on the scalp of a patient, brain activity can be recordednoninvasively. The variations of the EEG signals in time, e.g.,event-related EEG phenomena, may give insight into the neuralactivity [1]. For disorders like epilepsy, where the spatial loca-tions of malfunctioning brain regions need to be determined,EEG source analysis is required [2]. Indeed, the determinationof the origin of specific EEG waveforms helps neurologists to

Manuscript received June 15, 2010; revised September 27, 2010, December7, 2010, and January 12, 2011; accepted January 16, 2011. Date of publicationJanuary 20, 2011; date of current version April 20, 2011. This work was sup-ported by the Project GOA07/GOA/006 and Project IUAP-B/0784. The work ofG. Crevecoeur was supported by the Fonds voor Wetenschappelijk OnderzoekVlaanderen (FWO). Asterisk indicates corresponding author.

B. R. Yitembe and R. Van Keer are with the Department of Mathemati-cal Analysis, Ghent University, 9000 Ghent, Belgium (e-mail: [email protected]; [email protected]).

*G. Crevecoeur is with the Department of Electrical Energy, Systems &Automation, Ghent University, 9000 Ghent, Belgium (e-mail: [email protected]).

L. Dupre is with the Department of Electrical Energy, Systems & Automa-tion, Ghent University, 9000 Ghent, Belgium (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TBME.2011.2107740

pinpoint the origin of the epilepsy and to evaluate the patient forresective surgery [3].

EEG potentials can numerically be simulated through the useof a proper head model and neural source model. The quasi-static Maxwell’s equations need to be solved in this forwardproblem [4]. The solution of EEG inverse problem involvesthe determination of the neural source locations, starting fromEEG measurements. When coupling the measurements to a nu-merical method, inaccuracies in the neural source locations areintroduced. The deterioration in localization accuracy of EEGsource analysis is because of: 1) noise in measurements [5] and2) numerical modeling inaccuracies. The EEG localization er-rors due to 2) can be caused: a) by the use of a wrong sourcemodel [6], i.e., use of a single dipole instead of multiple dipolesresults in localization errors [7]; b) due to the use of a headmodel that is not accurate enough [8]; c) due to the misloca-tion of electrode positions [5]; and d) due to the use of wrongmaterial parameters, i.e., assumed conductivity values of pa-tient may differ significantly from the actual values and mayintroduce large localization errors [9], [10], and the assumptionof isotropic conductivities in the head model instead of actualanisotropic conductivity values may introduce large localizationerrors [11].

The method described in this paper aims at increasing thespatial localization accuracy by mitigating the propagation of theuncertain conductivity values to the inverse solutions. The brain-to-skull ratio of the conductivity is an important input parameterand is experimentally difficult to measure. In literature, largevariations in the conductivity values are observed [12]–[15]. Inthis paper, we assume that it varies between 1/60 and 1/9, whichcorresponds with the range of different measured values in theaforementioned papers.

In order to perform an accurate analysis of the presented novelnumerical scheme, the so-called reduced conductivity depen-dence (RCD) method, we assume that the neural source can berepresented by a single dipole or by a limited number of multipledipoles and that the electrode positions are accurate. Since this isa first step toward an increase of dipole localization accuracy forEEG source analysis, we use a widely used simple head modelgeometry: three-shell spherical head model [16]. In this way, weare able to perform sensitivity analysis in a computationally ef-ficient way and examine the efficiency of the proposed method.The RCD method introduces an alternative cost function, whichneeds to be minimized, and a selection procedure of the EEGelectrodes. These selected electrodes are least influenced by theuncertain conductivity values. Comparisons are made with the

0018-9294/$26.00 © 2011 IEEE

YITEMBE et al.: RCD METHOD FOR INCREASE OF DIPOLE LOCALIZATION ACCURACY IN THE EEG INVERSE PROBLEM 1431

Fig. 1. Spherical head model used for EEG forward and inverse problem.

widely used traditional least-squares minimization method. Theresults obtained in this paper can qualitatively be extended tomore accurate realistic head models.

II. EEG SOURCE ANALYSIS

A. Forward Problem

The forward problem computes the potentials at the elec-trodes, for a given electrical dipole location rd . The sphericalhead model is a widely used approximation [16], [17] of thehead, where it is represented by three spheres: the inner sphererepresents the brain, the intermediate layer represents the skull,and the outer layer represents the scalp. The skull-to-brain ratioof the conductivity X needs to be provided as well as the radiusof the different spheres. The conductivity of the brain σBrainand of the scalp σScalp is assumed to be equal. The different ra-dius can be obtained by fitting the spherical head model withinthe brain and skull compartments in the realistic head model.See Fig. 1, and for more details see [18]. The forward problemconsists in solving the Poisson’s equation [4]

∇ · (σ(r)∇V (r)) = ∇ · Ji(r) (1)

where Ji(r) is the impressed current density, σ(r) is the place-dependent conductivity determined by X , and V (r) is the place-dependent potential. In the following, rd = [rx, ry , rz ]T denotesthe dipole location and d denotes the 3-D strength of the dipole.

We fix σScalp ≡ σBrain = 0.33 S/m and σBrain is determinedby X . An analytical expression for the potential values canbe calculated using [16]. In this study, a standard configura-tion of m = 27 electrodes is used. For a given single dipolelocation rd and 3-D strength d, the electrical potential valuesVm (rd ,d) ∈ Rm×1 can be calculated at the predefined elec-trode positions. The potential values are a linear function of thedipole orientation

Vm (rd ,d) = L(rd) · d (2)

where L ∈ Rm×3 is the so-called lead field matrix.

B. Traditional Solution of EEG Inverse Problem

The aim of the EEG inverse problem is to recover the neuraldipole location r∗d and orientation d∗ that correspond the bestto the measured EEG potentials Vmeas ∈ Rm×1 . For a singledipole, this is carried out by minimizing a least-squares costfunction, the so-called relative residual energy (RRE) [4]

{r∗d ,d∗} = arg minrd ,d

RRE(rd ,d) (3)

with

RRE(rd ,d) =‖Vmeas − Vm (rd ,d)‖

‖Vmeas‖(4)

where ‖.‖ is the L2 norm with Vm the potentials obtained from(2). The above inverse problem is formulated to solve the EEGinverse problem at a single time instant and can be extended (formultiple time instances) in a spatiotemporal way, see (7).

The number of parameters in this least-squares cost functioncan be reduced by considering the optimal dipole components

dopt = L† · Vmeas (5)

where L† is the Moore–Penrose pseudoinverse of the lead fieldmatrix. Equation (4) becomes then, see, e.g., [4]

RRE(rd) =‖Vmeas − L(rd)L(rd)†Vmeas‖

‖Vmeas‖. (6)

The widely used Nelder–Mead simplex method is implementedhere to find the global minimum of the RRE in (3) (see, e.g., [18],[19]).

The solution of the EEG inverse problem for multiple dipoles(with number of dipoles less than the number of channel mea-surements) can be obtained using the recursively applied andprojected multiple signal classification (RAP-MUSIC) method-ology [7]. We denote here the so-called spatiotemporal datamatrix as

Fm = [L(rd,1), . . . ,L(rd,p)][dT1 , . . . ,dT

p ]T (7)

for p dipoles (with locations rd,i , i = 1, . . . , p). The 3 × n-matrix di (i = 1, . . . , p) represents the time course of the ithdipole moment and n is the number of samples registered bythe EEG. The RAP-MUSIC algorithm uses the separation of them-dimensional measurement space Fmeas into a signal Φsig andnoise Φn subspace, by calculating its autocorrelation matrix Rv .The singular value decomposition (SVD) of Rv can be writtenas follows:

Rv = E{FmeasFTmeas} (8)

= [ΦsigΦn ]

[Λsig 0

0 Λn

][ΦsigΦn ]T (9)

where Λsig and Λn are the diagonal matrices with the singu-lar values of Fmeas associated with the signal and the noisesubspace, respectively. The RAP-MUSIC algorithm uses a re-cursive procedure in which each source r∗d,k (k = 1, . . . , p) isfound as the global maximizer of a different cost function

r∗d,k = arg maxrd

(subcorr(Π⊥Ak −1

L(rd),Π⊥Ak −1

Φsig )1) (10)


where Π⊥Ak −1

= (I − Ak−1A†k−1) is the projection matrix con-

structed by Ak−1 , a matrix containing in each column the to-pographies of the already found k − 1 sources [7]. More de-tails with respect to the calculation of the subspace correlationsubcorr1 function can be found in [20]. The RAP-MUSIC algo-rithm may have some limitations [21], [22], but serves well thepurpose of the study presented in this paper.

III. RCD METHOD

For simplicity, we explain the method in the Section III-A–Cfor the case of single dipole localization. In Section III-D, weexplain what needs to be altered for the case of multiple dipolelocalization.

A. Description of the Method

The RCD method proposes: 1) an alternative cost functionthat needs to be minimized within the iterative minimizationmethod (see Section III-B) and 2) a selection procedure forthe EEG electrodes (see Section III-C). The main idea lies inselecting the set of electrodes that provide the most correctpossible information, i.e., electrodes that are minimally affectedby the unknown conductivity in the forward model evaluation.Indeed, depending on the location of the electrical dipole and itsorientation, some potentials are highly affected by X, and othersare not. The selection here is based on the sensitivity value ofthe potentials, i.e., the derivative of the analytical solution of (2)

W =∂Vm

∂X(11)

at a certain position rd , d. Only Ns potentials are selected forsolving the inverse problem. It is eventually also possible toselect the potentials with sensitivity values less or equal to acertain threshold α. The selection procedure needs to be per-formed in each iteration k of the minimization scheme, whichis in this case the Nelder–Mead simplex method. Different sub-sets of electrodes can be selected through this procedure. Weassume that the patient has a certain conductivity ratio X0 . Inthe following, we explain the basic steps of the proposed RCDmethod.

Step 1: Initial estimate r(0)d is evaluated in the forward model,

yielding the lead field matrix L(r(0)d ) and simulated

potential values Vm (r(0)d ) = L(r(0)

d )L(r(0)d )†Vmeas .

The start value is commonly provided by the user forminimization of the cost function. Here, we use asstart value always the center of the spheres. For theforward calculations, the assumed conductivity valueis used. Initialize k = 0.

Step 2: Calculate the sensitivity of the simulated potentialvalues to the conductivity for a certain conductivityratio X0

W(r(k)d ) =

∂(L(r(k)d )L†(r(k)

d )Vmeas)∂X

|X =X 0 (12)

in the kth iteration of the minimization procedure forfixed r(k)

d . In (12), (5) is used. In the case of the

spherical head model, W can be computed analyti-cally. When considering more complex realistic headmodels, this can be calculated by numerical finitedifferentiation.

Step 3: Selection of the Ns least sensitive electrodes, basedon the values of each electrode in (12). Potentials withlargest values of the sensitivity values are not consid-ered in the new approach for the EEG inverse problemdue to their high sensitivity to the conductivity. It isalso possible to set a certain threshold for sensitiv-ity values where a subset of potentials is formed andused in the inverse procedure. In order to be able tocompare simulated and measured potentials, the sameselection using the same index for each selected elec-trode is carried out on the measured EEG potentialsVmeas . We denote in the further discussion the se-lection operator as sel(·) that reduces the P ∈ Rm×1

to Q ∈ RNs ×1 when performing Q = sel(P). Leadfield matrices can also be selected on the same basis:from m × 3 to Ns × 3-D matrices. The followingnew sets of potential values in this kth iteration of theminimization procedure are obtained as

Sm = sel(Vm ) Smeas = sel(Vmeas). (13)

Also, a selection is carried out on the lead field matrix

M(r(k)d ) = sel(L(r(k)

d )) ∈ RNs ×3 . (14)

Step 4: Calculation of more correct dipole orientation usingthe selected lead field

d(k)opt = M†(r(k)

d )Smeas (15)

which is less affected by the uncertain conductivityratio than L†(r(k)

d )Vmeas .

Step 5: Calculation of RCD cost function: c(k) = RCD(r(k)d ,

d(k)opt). See Section III-B.

Step 6: Based on the value of the cost function, the next iterater(k+1)

d can be calculated. If the termination criteria ofthe minimization procedure are met, i.e., c(k) reachestolerance ε, then stop the algorithm. Otherwise, up-date k = k + 1 and go to step 2.

B. Cost Function of the RCD Method

If we define the cost function in step 5 of the iterative proce-dure as the RRE of Smeas and Sm , then the solution of the RCDmethod converges to the same solution

r∗d = arg minrd

‖Smeas − M(rd)M(rd)†Smeas‖‖Smeas‖

(16)

as the traditional minimization method, which minimizes (6).The only difference is that for the same tolerance ε, differentsolutions will be obtained, depending on the selected potentials.In the theoretical limit k → ∞, the same set of solutions willbe obtained as in the traditional methodology with RRE. In thecase of no noise, this corresponds with ε ≡ 0.

For an assumed value of the conductivity X0 , with simulatedEEG potential Vm (rd ,X0), the EEG potential Vm (rd ,X) for


Fig. 2. Sensitivity as a function of the difference in potentials for rd at CS3,with assumed X0 = 1/13 and actual X = 1/9.

different conductivity X can be approximated by a first-orderTaylor expansion

Vm,T (rd ,X)=Vm (rd ,X0) + (X −X0)∂Vm (rd ,X0)

∂X|X =X 0 .

(17)This potential needs to approximate at the actual dipole locationr∗d the measured EEG potential, for which the patient understudy has the actual conductivity ratio X

Vm,T (rd , X) ≈ Vmeas . (18)

Remark that ≈ is used because of the presence of noise inmeasurements. Following this idea, we need to minimize thefollowing cost function:

‖Vmeas − Vm (rd ,X0) − (X − X0)∂Vm (rd ,X0)

∂X|X =X 0 ‖.

(19)However, the actual conductivity ratio X in the brain of thepatient is not known, but we can approximate the Taylor co-efficient α = X − X0 by fitting Vmeas − Vm (rd ,X0) linearlyto the sensitivity (12). This is the reason why the followingalternative cost function is defined:

RCD(rd) =‖Smeas − Sm (rd ,X0) − αQ(rd)‖

‖Smeas‖(20)

with Q = sel(W(rd)). This cost function needs to be mini-mized for the recovery of the dipole location and orientation sothat instead of (3)

{rd , d} = arg minrd ,d

RCD(rd) (21)

has to be carried out. We remark that higher order terms can beincluded in the RCD cost function. For example, the second-order term, a fit needs to be carried out in a plane with thefirst-order and second-order derivative to Smeas − Sm (rd ,X0).Remark that if X → X0 , then the definition of RCD → RRE.

In order to illustrate the choice of the above cost function, weshow the sensitivity in Fig. 2 [as defined in (12)]

∂Vm (rd)∂X

|X =X 0 =∂(L(rd)L†(rd)Vmeas)

∂X|X =X 0 (22)

as a function of the difference Vmeas − Vm (rd ,X0). We usehere as Vmeas the potentials that correspond with the same lo-cation but with actual conductivity ratio X �= X0 : Vmeas =

TABLE ICASE STUDIES (CSS) OF SEVERAL SINGLE DIPOLES LOCATED AT DIFFERENT

REGIONS. THE CENTER OF THE HEAD MODEL IS REFERENCED AS r = [0, 0, 0]

Fig. 3. Sensitivity as a function of the difference in potentials for rd at CS3and rd is approximately 2 mm distant from rd , with X0 = 1/13 and X = 1/9.

Vm (rd ,d, X). The location rd corresponds to the locationCS3, given in Table I, and its orientation is randomly cho-sen (d = [0.5 0.4 0.7]T ). As observed in Fig. 2, the Taylorcoefficient (X − X0) can be fit. As a comparison, we show,in Fig. 3, the sensitivity evaluated at a different locationrd �= rd : (∂Vm (rd))/∂X|X =X 0 versus the difference Vmeas −Vm (rd ,X0), where Vmeas = Vm (rd , X). In Fig. 3, we showthe absolute values on the x- and y-axes. We should note herethat rd is approximately 2 mm away from rd . Notice that in thecase of real measured potentials with noise, the difference canbecome larger and that the residual of the linear fit becomes alsolarger. Additional localization errors will be induced.

C. Selection Procedure of the RCD Method

The RCD method offers the possibility to choose a certainnumber of electrodes out of the given electrode configurationsetup. As said already, the selection is based on sensitivity val-ues. In this paper, the choice of the number of potentials (Ns) tobe used in the RCD method is fixed. The smallest value of Ns

out of the standard configuration that could be used for the RCDmethod is 7, otherwise the inverse problem becomes ill-posed.

For a dipole located in the region CS3, we calculated the po-tential values Vm for the m = 27 different electrodes and inves-tigated the sensitivity W [see (11)] in Fig. 4. This figure showsthat the potentials measured at the different electrodes are notequally influenced by the uncertain conductivity ratio. Indeed,potentials measured at certain electrodes have a high sensitivityto the conductivity ratio, while other potentials have a relativelylow sensitivity. In addition, Fig. 4 shows the set of sensors thatneeds to be selected when solving the inverse problem. Elec-trodes with high sensitivity values will give rise to a large errorin the dipole localization. This is the reason why we do eliminate


Fig. 4. Sensitivity as a function of the conductivity ratio for dipole located atposition CS3. As shown, some electrodes are very sensitive to a conductivitychange, while others have a limited sensitivity.

these potentials in the minimization procedure. This figure alsoshows that a selection of a proper set of electrodes (electrodeswith less conductivity influence) can be carried out. Throughthe use of a suitable selected set of potentials, the dipole orien-tation (15) can be determined in a better way. Furthermore, amore correct comparison between measured and simulated po-tentials can be performed. Indeed, it is possible to “weight” thepotential values of simulated and measured potentials so thatcost function 6 becomes

RREw (rd) =‖KVmeas − KL(rd)L(rd)†Vmeas‖

‖KVmeas‖(23)

where K is a weighting matrix that is diagonal. The ith diagonalelement Ki,i needs to be reciprocal to the sensitivity of the i-thpotential. In the following, we use Ki,i = 1, if the ith potentialis among the Ns least sensitive potentials with (Ns < N ). Oth-erwise, Ki,i = 0. If a weighting matrix is used, the followingcost function should be used [see also (20)]

RCDw (rd) =‖KVmeas − KVm (rd ,X0) − αKQ(rd)‖

‖KVmeas‖.

(24)

When using the approach Ki,i = 1 or 0, has an advantage:the dipole orientation (15) can de determined in a better way,while this is not possible when using continuous Ki,i values.Furthermore, a more correct comparison between measured andsimulated potentials can be performed.

D. RCD Methodology for Multiple Dipoles

The basic principles explained in previous sections for singledipole localization (minimization of RRE) can also be ap-plied for multiple dipole localization. Since the MUSIC al-gorithm is a scanning method and the RAP-MUSIC performssequentially p different maximizations or minimization of costfunction −subcorr(Π⊥

Ak −1L(rd),Π⊥

Ak −1Φsig )1 [see (10)], the

RAP-MUSIC algorithm is a suitable algorithm for the RCDmethodology. For a single dipole, the cost function (6) was al-tered to (21), while for the recovery of multiple dipoles, the costfunction in (10) can be altered using the RCD methodology. An-other reason for the use of the RAP-MUSIC algorithm insteadof the MUSIC algorithm is because we can still select a low

number of electrodes in the methodology. Indeed, the numberof unknowns when performing a single maximization in (10) is6 (3 for location and 3 for orientation) so that the inverse prob-lem is not ill-posed when selecting, e.g., 10 electrodes (contraryto the possible use of RCD within MUSIC).

When incorporating the RCD methodology within theRAP-MUSIC framework, the steps 1–6 given in Section III-Acan be executed in a similar way. However, the sensitivity of thesimulated potential values to the conductivity in step 2 needsto be altered. We consider the sensitivity to the conductivityratio of the principal vector U ∈ Rm×1 that is associated withthe principal angle of the subspace correlation function. Thisprincipal vector can be calculated following [20], and dependson the kth iteration of the minimization procedure upon Φs andL(r(k)

d ) (and consequently also upon the conductivity ratio X).Therefore, instead of using (12), the following sensitivity-to-conductivity ratio is calculated in each iteration

Y(r(k)d ) =

∂U(r(k)d )

∂X|X =X 0 . (25)

This can be calculated in a finite-difference way. Selection ofNs electrodes with selection operator sel(·) is carried out on thebasis of (25), similar to step 3 in Section III-A. A fitting needsto be carried out using data vectors U and Y, resulting in fittingconstant α.

Instead of cost function (20), a similar cost function is evalu-ated in step 5:

subcorr(sel(Π⊥Ak −1

(L(rd) + αW(rd))), sel(Π⊥Ak −1

Φsig ))1(26)

which is based on (10). Following the basic principle of the RCDmethod, a first-order extended forward model is used: L(rd) +αW(rd) with W ∈ Rm×3 defined here as the sensitivity of thelead field matrix to the conductivity ratio

W =∂L(rd)

∂X|X =X 0 . (27)

Remark that in the RAP-MUSIC framework, di (i = 1, . . . , p)defined in (7) can be determined in a similar way as (5).

IV. RESULTS

A. Simulation Setup

The efficiency of the RCD method is investigated by per-forming numerical experiments. For the localization of singledipoles, no temporal data are generated. Starting from knowndipole location rd , d and actual conductivity ratio X , we com-pute the EEG potentials V. To these potentials, Gaussian noiseN is added in order to simulate real measured EEG potentialsVmeas = V + N. See Fig. 5. The white zero mean Gaussiannoise with standard deviation Σ has a noise level defined asn = Σ/(Vrms), where Vrms is the root mean square of Vmeas .The numerical experiments consist in solving the EEG inverseproblem using an assumed conductivity ratio X0 . We estimatethe dipole position r∗d using the traditional method (3) and theposition rd using the RCD method (21). Dipole location errorswill be introduced in both methods because of noise (n > 0) and


Fig. 5. Numerical experiments for investigating the efficiency of the method-ology. Noise set and difference between actual and assumed conductivity intro-duce dipole location errors.

TABLE IIDIPOLE LOCATIONS AT DIFFERENT REGIONS IN THE CASE OF MULTIPLE

DIPOLE LOCALIZATION. CENTER OF THE HEAD MODEL IS REFERENCED

AS r = [0, 0, 0]

assumed conductivity ratio values different from actual conduc-tivity values (X0 �= X). The accuracy of both methods is deter-mined by the dipole position error: DPE = ‖rd − rd‖. Noticethat the start value r(0)

d for the minimization of (3) and (21)is always chosen as the center of the brain. In this paper, thescalp electrodes are placed following the 10–20 internationalsystem [23], with three electrodes on each of the interior tem-poral regions adding up to m = 27 electrodes.

In order to test the RCD algorithm, we perform numericalexperiments on various regions of the spherical head model.Four case studies (CSs) are taken within the spherical headmodel: the center (CS1, CS2), the interior (CS3), and the edge(CS4) (see Table I). They give an overview of the geometricalmodel and are therefore chosen for this purpose.

The simulation setup for the recovery of multiple dipolesconsists in the recovery of p = 2, p = 3, and p = 5 differentdipoles. For the location of these dipoles, see Table II. In thecase of p = 2, we assume that the first dipole is located atDL1, which represents an epileptic spike of 0.2 s with onsetat 0.4 s, and that the second dipole is located at DL2 withrhythmic activity (sinusoidal waveform at 10 Hz). For the casep = 3, the third dipole is located at DL3 with rhythmic activity(cosinusoidal waveform at 10 Hz). For p = 5, a dipole at DL4(sinusoidal waveform at 5 Hz) and a dipole located at DL5(cosinusoidal waveform at 5 Hz) are added. Using the forwardmodel, a spatiotemporal matrix (7) was generated with a certainconductivity ratio value X . To this matrix Fm , Gaussian noiseN was also added with varying noise levels. Fig. 6 shows anexample of the EEG data (1 second) used during the simulationexperiments. The total number of EEG electrodes is here alsom = 27.

B. Dipole Localization Accuracy of the TraditionalLeast-Squares Minimization Method

As mentioned in section I, the sources of localization errorsin EEG are the noise in measurements and the uncertaintiesin the forward numerical model. The only uncertainty that is

Fig. 6. Sample of average referenced EEG data. This EEG data results fromp = 3 dipoles (located at DL1, DL2, and DL3).

Fig. 7. Dipole position error for CS4 due to noise only in measurements usingthe traditional method. No uncertainties are assumed in the forward model:X = X0 .

investigated in this paper is the assumed conductivity ratio val-ues X0 . Fig. 7 shows the DPEs introduced by noise n in themeasurements with X = X0 .

When we assume that no noise is available in the measure-ments, but that the forward model has uncertain conductivityvalues, then large dipole position errors can be introduced. Asan example, see Fig. 8. We observe that the use of the uncertainconductivity ratio value may introduce errors, which can evenbe larger than the errors due to noise only in measurements.

C. Single Dipole Localization Accuracy of the RCD Method

In a first stage, we compare the results of the traditionalmethod with the proposed RCD method using numerical exper-iments with no noise n = 0 available in the potentials. Fig. 8shows the localization errors for the four different CSs whenX0 �= X . In this figure, the alternative cost function is imple-mented without selecting the EEG electrodes: Ns = m, wherem = 27. The results show an improvement in single dipole lo-calization accuracy when using the RCD method.

When selection with Ns < m is carried out, the localizationerror decreases. This, because the highly sensitive potentialsintroduce a large localization error.

In a second stage, we add Gaussian noise n > 0 to the simu-lated potentials with actual conductivity ratio X . Fig. 9 showsthe localization errors for CS4 when noise n = 0.025 is availablein measurements for the different X0 . Note that the error of theRCD method is the same as the traditional RRE minimizationmethod when X0 is close to X . For X �= X0 , the localizationerror when using the RCD method is decreased. With and with-out noise, we have better localization accuracy. The efficiencyof the proposed method for different noise levels is shown in


Fig. 8. Dipole position error for the different CSs using the traditional least-squares minimization of the RRE and the RCD method. The actual conductivityratio of the simulated Vm eas is chosen as X = 1/16.

Fig. 10 with X = 1/16 and X0 = 1/40. Less procentual increaseof accuracy when using the RCD method compared to the RREmethod is obtained for increasing noise levels.

D. Multiple Dipole Localization Accuracy of the RCD Method

We applied the methodology explained in Section III-D forthe recovery of multiple dipoles. We applied the RCD methodon spatiotemporal EEG data, as illustrated in Fig. 6. Figs. 11 and12 depict the dipole localization error for, respectively, p = 2and p = 3 dipoles when using the traditional RAP-MUSIC costfunction (denoted as “no selection”) and when using the RCDmethodology during the maximization of cost function (26) inthe no noise case. The methods reach, in Figs. 11 and 12, the bestaccuracy around X0 = 0.04 because in this simulation study, the

Fig. 9. Dipole position error for CS4 with noise (n = 0.025) when usingthe traditional least-squares minimization of the RRE and the RCD method.Potentials are generated with X = 1/16. No selection of potentials is carriedout Ns = 27.

Fig. 10. Dipole position error for CS4 with X = 1/16 and X0 = 1/40 fordifferent noise levels. Number of selected potentials is Ns = 10.

Fig. 11. Dipole position errors for dipoles located at DL1 and DL2 whenassuming different conductivity ratios. The EEG data were generated with actualconductivity ratio X = 1/24 and no noise was added to the EEG data. The twoupper graphs are the errors when using the traditional RAP-MUSIC method(No selection), while the two lower graphs show the errors when using theRAP-MUSIC method with the use of RCD methodology. The number of selectedelectrodes is here Ns = 12.

actual conductivity ratio is X = 1/24. For clarity reasons, weshow the errors within the range X0 = 0.015 − 0.04.

We also added noise N to the generated EEG data with vary-ing noise levels n. Fig. 13 shows the total dipole localizationerror, i.e.,

∑pj=1 DPEj versus the noise level in the case of

p = 3. DPEj is the dipole position error for the jth dipole, andthey were calculated as the average of the results of 100 EEGdata samples (with fixed noise level). The same results wereobtained when recovering p = 5 different dipoles (see Fig. 14).


Fig. 12. Dipole position errors for dipoles located at DL1, DL2, and DL3when assuming different conductivity ratios. The EEG data were generatedwith actual conductivity ratio X = 1/24 and no noise was added. The numberof selected electrodes is here Ns = 12.

Fig. 13. Total dipole position error when recovering p = 3 dipoles (DL1,DL2, and DL3). The assumed conductivity ratio is here 1/60, while the actualconductivity ratio of the EEG data is about 1/24. The number of selected elec-trodes is Ns = 12. Each data point is calculated as the average of 100 EEGsample with certain noise level.

Fig. 14. Total dipole position error when recovering p = 5 dipoles (DL1,DL2, DL3, DL4, and DL5). The assumed conductivity ratio is here 1/60, whilethe actual conductivity ratio of the EEG data is about 1/24. The number ofselected electrodes is Ns = 12. Each data point is calculated as the average of100 EEG sample with certain noise level.

V. DISCUSSION

A. Influence of Ns on Accuracy

Fig. 15 shows the localization error in the single dipole casewhen selecting a fixed number of potentials in the iterativeprocedure of the RCD method. We observe a global behavior ofdecrease in dipole position error. However, between Ns = 14and Ns = 22, an increase of error is observed. This is becausethe minimization of the RCD cost function does not result ina global optimum. When using other start values, decrease of

Fig. 15. Dipole position error for CS3 with assumed conductivity ratio X0 =1/30 and actual conductivity ratio X = 1/16.

Fig. 16. Dipole position error for CS3 with noise level n = 0.2. Assumedconductivity ratio X0 = 1/30 and actual conductivity ratio X = 1/16.

error between Ns = 14 and Ns = 22 can be observed. Thisis the main drawback of the RCD method: global minimizationtechniques are needed. Multistart Nelder–Mead simplex methodis an alternative. Notice that it is possible to not “weight” theelectrode channels in a black and white way, as proposed inSection III-A, but that weighting factors can also be used foreach electrode channel. Such an approach, however, needs tobe further elaborated. Also, the selection criterion, in this paperthe selection of a fixed number of selected potentials Ns , issuboptimal and the use of, e.g., (∂V )/∂X ≤ α with selectionthreshold α, can be used. In this way, the number of selectedpotentials will be dynamic in each iteration. A mathematicalanalysis is needed for understanding the working of such amethodology.

Fig. 16 shows the dipole position error when noise n = 0.2is available in Vmeas . We observe again that the efficiency ofthe RCD method decreases when noise is available in the mea-surements. However, there is still an increase of accuracy with1.4 mm. The reason for having the lower efficiency is becausethe fitting procedure in the RCD cost function is not so accurateanymore. Remark that determining the actual conductivity ratioX is difficult because dipole localization errors are introducedand because of the noise in the measurements.

In the case of using the RCD methodology for the recoveryof multiple dipoles, the same conclusion can be drawn withrespect to the influence of Ns on the accuracy. Fig. 17 showsthe decrease in total dipole position error when decreasing thenumber of selected electrodes Ns . EEG data (see Fig. 6) were


Fig. 17. Total dipole position error with assumed conductivity ratio 1/60 andactual conductivity ratio is about 1/24.

Fig. 18. Indices of selected potentials during minimization of RCD cost func-tion with X0 = 1/20 and X = 1/16. Recovery of dipole CS2 with Ns = 10electrodes during the minimization of the RCD cost function.

generated for p = 3 dipoles (DL1, DL2, and DL3) with noiselevel n = 0.3.

B. Convergence History

When minimizing in the single dipole case, the RCD costfunction given by (24), only a subset of potentials (Ns out of Npotentials) is used. Fig. 18 shows the indices in each kth iterationof the minimization process (21). Indeed, in step 3 of the RCDmethodology, only a specific subset of potentials is used, see theselection operator sel(·) in (13) and (14). Here, in each iteration,Ns = 10 potentials are selected in each iteration, which corre-spond with the potentials with lowest sensitivity-to-conductivityratio X (see also Fig. 4). We observe at the beginning of theminimization process (k < 40), many variations in the set of se-lected potentials, while near the end of the minimization process(k > 100), the set of selected potentials stays relatively fixed.This is because in the beginning, r(k)

d may vary a lot, while near

the end of the minimization process, r(k)d reaches slowly toward

the optimal solution with the RCD cost function converging tothe stopping criteria ε. W(r(k)

d ) [see (8)], will not vary muchin these iterations. Notice that the whole set of 27 potentialsis employed in the minimization of the RCD cost function, butthat only a subset of potentials is used in each kth iteration.The same behavior of convergence history is observed whenlocalizing multiple dipoles.

Fig. 19 shows the value of the averaged referenced signalsrecorded during the minimization of the RCD cost function. This

Fig. 19. Average referenced amplitude of signals, calculated during each kthiteration of the minimization procedure with ×: the values of the signals simu-lated at the 27 electrodes, and ©: the values of the selected potentials at the Ns

electrodes. This minimization is for CS1, X0 = 1/40, X = 1/16, and Ns = 10.

Fig. 20. Average referenced sensitivity of the whole set (×) and of the selectedset of potentials (©) to the location x: (∂V )/∂x, calculated during each kthiteration of the minimization procedure. This minimization is for CS1, X0 =1/40, X = 1/16, and Ns = 10.

figure illustrates the selected potentialsSm that are selected fromthe Vm [see (13) in step 3 of the RCD method]. It is clear thatnot only low-amplitude voltages are selected, but also potentialswith high amplitude are selected. The encircled values depictthe selected potentials. The selected potential values are not onlynear the V = 0, but also electrodes are selected that are near themaximum value (or minimum value).

Moreover, Fig. 20 shows that the selected potentials (withlowest sensitivity to the conductivity ratio) do not always displaythe lowest sensitivity to the location of the dipole. The encircledvalues depict the value of (∂Sm )/∂x (out of the (∂Vm )/∂x val-ues) in each iteration of the minimization. The same is observedfor (∂Vm )/∂y and (∂Vm )/∂z. In future work, investigationswill be made for improving the RCD methodology by incorpo-rating the sensitivity of the potential values to the dipole location(∂Vm )/∂rd in each iteration of the minimization. In this way,it may be possible to increase the efficiency of the method whendealing with noisy data.

Fig. 21 shows the convergence history of the RCD method fora single dipole. The RRE cost function was also recorded duringthe minimization of the RCD method. The iterative procedureneeds approximately 50–100 forward function evaluations inorder to minimize the RCD cost function. The minimization iscarried out using the Nelder–Mead simplex method. Remarkthat other minimization procedures can be implemented. The


Fig. 21. Convergence history when minimizing the RCD cost function fordipole at CS3 with conductivity ratio X = 1/40, starting from potentials thatare generated with conductivity ratio X = 1/25. During the minimization of theRCD cost function, the value of RRE is given.

Fig. 22. Dipole position error for p = 1 dipole located at CS4 using EEG dataof Fig. 10.

RCD method can be implemented in any possible iterative min-imization procedure. The dipole location that corresponds withthe minimum of the RCD method gives a better solution, sinceits error is lower, compared to the one of the RRE cost function.We deduce from this figure that the RRE value is large whenwe are close to the accurate solution. As illustrated in Fig. 15,an improvement of the localization accuracy is also obtainedwith the selection of potentials. Moreover, the minimal value ofthe RCD method does not correspond with the minimal valueof the RRE method. The selection of the potentials, with lesssensitivity value in the RCD method, produces a cost function(24) that is by definition lower than the one of the traditionalmethod (4) in the L2 norm. The aforementioned can also beconcluded when recovering multiple dipoles.

C. Influence of Noise on Accuracy

In Fig. 10, we observed that for the recovery of a single dipolethe efficiency of the RCD method [with cost function (20)] isdrastically reduced compared to the no-noise case. However,when we compare these results for single dipole reconstructionwith Figs. 13 and 14 for multiple dipole reconstruction, we ob-serve that the RCD with cost function (26) is more robust tonoise. The first reason for this better performance is that weare using SVD where the removal of components with smallereigenvalues reduces the effect of noise. The second underlyingreason for this is that the fitting constant α is much better ap-proximated when using as data U and Y (see Section III-D),since the RAP-MUSIC and the RCD with RAP-MUSIC have

procentually the same relative increase in total dipole positionerror due to noise. The principal vectors [and corresponding sen-sitivity (25)] are less affected by noise. Indeed, we applied theRCD with (26) onto the localization of a single dipole (CS4) andadded several levels of noise (same data as in Fig. 10). Fig. 22shows indeed a more robust behavior of the RCD method towardnoise when using the subspace correlation function (26).

VI. CONCLUSION

This paper proposes a method that decreases the localizationerror introduced by uncertain conductivity values when solvingthe inverse problem. This is carried out by reformulating thecost function that needs to be minimized and by selecting aproper set of EEG electrodes that are minimally influenced bythe conductivity values. The results show that the EEG inverseproblem can be solved with considerably improved quality, ascompared to the traditional inverse solutions.

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Authors’ photographs and biographies not available at the time of publication.

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