EEG Dipole Source Localization in HemisphericalHarmonics DomainAmita Giri, Lalan Kumar and Tapan Gandhi

Indian Institute of Technology, DelhiE-mail: eez178183, lkumar, tgandhi@ee.iitd.ac.in

Abstract—Dipole source localization problem is addressed inhemispherical harmonics (HSH) domain. Human head is approx-imated by spherical head model. Hence, spherical harmonics(SH) basis functions have been natural choice for EEG sourcereconstruction and localization. However, sensor placed overscalp to acquire EEG signal, assume the shape of a hemisphere.Hemispherical harmonics basis functions will more appropriatelyrepresent the data sampled over hemisphere. In this paper, theforward data model is formulated in HSH domain. Optimalarray processing methods for source localization such as MUltipleSIgnal Classification (MUSIC) and minimum variance distortion-less response (MVDR) are proposed in HSH domain. Variousexperiments have been presented to illustrate the theory.

I. INTRODUCTION

Dipole source localization (DSL) has been an active area ofresearch in the recent years because of clinical and researchapplications [1, 2, 3]. DSL is done using electroencephalogram(EEG) or magnetoencephalogram (MEG) signals. EEG signalgeneration is modeled using equivalent charge, equivalentpotential and equivalent current dipole. The equivalent currentdipole (ECD) model is used widely as it models ion flow fromone neuron (current source) to the other neuron (sink). Currentsource and sink constitute the current dipole in brain. Thecurrent dipole is parameterized by its magnitude, location andorientation. DSL involves estimation of these dipole parame-ters. The DSL problem is formulated using underdeterminedand overdetermined approach, referred as dipole imaging anddipole fitting respectively. Underdetermined source modelsgive distributed solutions all over the brain. Overdeterminedapproach to DSL gives compact representation of distributedsources in a clustered sense, thus can be used to repre-sent concisely the source distribution. Various approaches tooverdetermined DSL problem have been proposed that includemodels for one dipole [4], multiple dipoles [5], Soft computing[6], likelihood estimation with fMRI data [7, 8], PCA fitting[9], artificial neural network [10] and MUSIC [11]. All theseapproaches make use of spatial domain signal processing.

Spherical harmonics domain processing has received sig-nificant attention for acoustic source localization and beam-forming in the recent years due to ease of array processingin SH domain with no spatial ambiguity [12, 13]. A spher-ical sensor array is utilized for this purpose. In literature,the human head is approximated by spherical head model[14]. Hence, spherical harmonics basis functions have beennatural choice for EEG source reconstruction and localization[15, 16, 17]. Spherical harmonics were utilized in spatial

Fig. 1: Typical head model with projection of sensorpositions over hemisphere.

filtering of MEG multichannel measurements to any user-specified spherical region of interest (ROI) inside the head[18]. Spherical harmonics in general, are utilized to repre-sent function defined over entire sphere. For DSL, the EEGsignal is acquired using sensors placed over scalp. Spatialsampling of EEG signal over scalp can be approximated asa hemisphere as shown in Fig. 1. Accurate representationof data over hemisphere by SH requires more number ofSH coefficients due to discontinuities at the boundary of thehemisphere [19, 20]. Hence, hemispherical harmonics basisfunction will more appropriately represent the data sampledover hemisphere. HSH basis functions are utilized in [19]for hemispherical radiance function representation and in [20]for acoustic source localization. In this paper, overdeterminedapproach based on hemispherical harmonics MUSIC algorithmis explored for DSL. The novelty of work is in utilization ofhemispherical harmonics domain processing for DSL.

II. SPATIO-TEMPORAL FORWARD DATA MODEL

In this Section, spatio-temporal data model is derived usinginfinite homogeneous isotropic conductor (IHIC) model. Theco-ordinate system utilized herein, assumes origin at the centerof head. I sensors are utilized to measure the potential overhead. The pth source location is denoted by −→rp = (rp, θp, ϕp)

T

where rp is distance from a source to the origin, θp is elevationangle measured downward from positive z axis, and ϕp isthe azimuth angle measured anticlockwise from positive xaxis. Similarly, the ith sensor location is given by −→ri =(ri, θi, ϕi)

T . The DSL problem under consideration, utilizesECD model where each dipole is parameterized by its position

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Fig. 2: The dipole parameters.

~rp = (xp, yp, zp)T and dipole moment ~D = (DpxDpyDpz)

T

as shown in Fig. 2.In the IHIC model, the brain medium is assumed to be

isotropic and homogeneous with constant conductivity σ toavoid the reflection effects [14]. At a given instant, P regionsare considered to be active in the brain medium. The electricpotential measured at a point −→ri on scalp generated by the pthactive region is given by the Poisson’s equation as [21]

Vp(−→ri , t) =

1

4πσ

∫Sp

−→Jv(t).

(−→ri −−→rpv)|(−→ri −−→rpv)|3

dv, (1)

where the integration is over a volume Sp containing the vthdipole located at −→rpv position vector.

−→Jv is the current dipole

density for a dipole at elemental volume dv. In ECD model,the dipoles in volume Sp is characterized by a equivalentcurrent dipole moment given by

−→Dp(t) =

∫Sp

−→Jv(t)dv. Hence,

the time varying potential at sensor location −→ri due to the pthequivalent dipole can be written as

Vp(−→ri , t) =

−→Dp(t).(

−→ri −−→rp )

4πσ|(−→ri −−→rp )|3(2)

where −→rp is the mean dipole location vector of dipoles activein the volume Sp. For the case of dipoles with fixed locationand orientation, the total potential at the ith sensor for P suchdipoles is given by

V (−→ri , t) =P∑p=1

Vp(−→ri , t) (3)

Utilizing (2) and (3), the spatio-temporal dipole data modelfor I sensors and Ns time samples, can be written in matrixform as

VI×Ns= GI×3PD3P×Ns

. (4)

where, G is called lead field matrix, given by

G =[G1 G2 · · · GP

](5)

with

Gp =[−→b p,1

−→b p,2 · · ·

−→b p,I

]T(6)

−→b p,i =

(−→ri −−→rp )

4πσ|(−→ri −−→rp )|3, (7)

and D is dipole moment matrix given by

D =[D1 D2 · · · DP

]T(8)

with

Dp =[Dpx Dpy Dpz

]T(9)

It is to note that−→b p,i is 1× 3 vector and each of Dpx, Dpy ,

Dpz is 1×Ns dipole moment time series vectors in X , Y , Zdirections. Under fixed dipole assumption, the dipole momentmatrix in (4), can further be decomposed as

D3P×Ns= M3P×PSP×Ns

(10)

where M represents unit orientation moments and S momentintensity matrix. Substituting this in (4), the final spatio-temporal data model can be written as

VI×Ns= AI×PSP×Ns

(11)

where

AI×P = GI×3PM3P×P . (12)

The resultant data model separates time invariant (location andorientation) and time variant portions (intensity) parameters ofthe dipole. Which facilitate the application of subspace basedmethods for source localization.

III. HEMISPHERICAL HARMONICS DOMAIN DATA MODEL

The spatio-temporal data model in (11) is reformulatedherein in hemispherical harmonics domain. The hemisphericalharmonics domain processing is computationally more effi-cient due to dimensionality reduction. In the ensuing Section,spherical harmonics decomposition of the measured potentialis introduced first.

A. Real Spherical Harmonics TransformDipole source localization is performed using discrete time

domain EEG. As the EEG potential signals (3) are real,we introduce the real spherical harmonics transform (SHT)defined on the sphere first. The real SHT of discrete timepotential measured over scalp ∈ (R,Ω) is given by

Vnm(t) =

∫Ω∈S2

V (t,Ω)[Y mn (Ω)]dΩ, (13)

where Y mn (Ω) is real valued spherical harmonics [22] of ordern and degree m, defined as

Y mn (θ, φ) =

(−1)|m|

√2Km

n sin(|m|φ)P |m|n (cos θ) : m < 0

(−1)|m|√2Km

n cos(mφ)Pmn (cos θ) : m > 0

K0nP

0n(cos θ) : m = 0

(14)

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Proceedings, APSIPA Annual Summit and Conference 2018 12-15 November 2018, Hawaii

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

(a) H00

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

(b) H−11

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

(c) H01

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

(d) H11

−0.5

0

0.5

(e) H−22

−0.5

0

0.5

(f) H−12

−0.4

−0.2

0

0.2

0.4

0.6

0.8

(g) H02

−0.5

0

0.5

(h) H12

−0.5

0

0.5

(i) H22

Fig. 3: Hemispherical harmonics basis upto order 2, (a)n = 0, (b) (c) (d) n = 1, (e) (f) (g) (h) (i) n = 2.

where Pmn (cos θ) is the Associated Legendre Polynomial(ALPs). Replacing cos θ by x for simplicity, the ALPs aregiven as

Pmn (x) =

(−1)m(1− x2)m/2

dm

dxmPn(x) : m ≥ 0

(−1)m(n−m)!

(n+m)!P|m|n (x) : m < 0

(15)

where Pn(x) is unassociated Legendre polynomials expressedas

Pn(x) =1

2nn!

dn

dxn(x2 − 1)n. (16)

The order n takes value from [0,∞) while m varies from[−n, n], |.| denotes the absolute value of (.), Km

n is thenormalization value given by

Kmn =

√(2n+ 1)(n− | m |)!

4π(n+ | m |)!. (17)

The associated Legendre polynomials for different order nand same degree m are orthogonal over [-1,1] with weightingfunction as 1 [19]. The orthogonality relation is given by∫ 1

−1

Pmn (x)Pmn′ (x)dx =

2 (n+m) !

(2n+ 1) (n−m) !δnn′ . (18)

The inverse real SHT is given by

V (t,Ω) =∞∑n=0

n∑m=−n

Vnm(t)Y mn (Ω) (19)

B. Hemispherical Harmonic Decomposition of EEG Signal

The orthogonality of Pmn (x) is over [-1,1] as indicatedin (18). The range [-1,1] is due to the fact that the ele-vation angle θ, in Pmn (cos θ) takes value in [0, π]. As, theEEG measurement is available over hemispherical scalp, theelevation of the sensor must lie in [0,π/2] or equivalently,x ∈ [0, 1]. Hence, a new set of orthogonal associated Legndrepolynomials is required. This is achieved by shifting theassociated legendre polynomials. In general, if the polynomialsPnm(x) are orthogonal over [a, b], with w(x) as a weightingfunction, then the polynomials Pnm(q1x + q2) where q1 6= 0are orthogonal over an interval [a−q2q1

, b−q2q1] with w(q1x+q2)

as a weighting function [23]. The linear transformation of xto 2x− 1 in (18) gives the shifted ALPs expressed as

Pmn (x) = Pmn (2x− 1) (20)

The shifted ALPs are orthogonal over [0, 1] with weightfunction as 1. The orthogonal relation is now expressed as∫ 1

0

Pmn (x)Pmn′ (x)dx =

∫ 1

0

Pmn (2x− 1)Pmn′ (2x− 1)dx

=(n+m)!

(2n+ 1)(n−m)!δnn′ . (21)

Just as ALPs are used to construct SH basis functions,shifted ALPs are utilized herein to construct a HSH basisfunctions orthogonal over [0, π2 ] × [0, 2π]. The real valuedhemispherical harmonics basis functions Hm

n (θ, φ) can nowbe expressed as:

Hmn (θ, φ) =

(−1)|m|

√2Km

n sin(|m|φ)P |m|n (cos θ) : m < 0

(−1)|m|√2Km

n cos(mφ)Pmn (cos θ) : m > 0

K0nP

0n(cos θ) : m = 0

(22)

where Kmn is the normalization value expressed as

Kmn =

√(2n+ 1)(n− | m |)!

2π(n+ | m |)!. (23)

2D plots of HSH basis functions upto second order are shownin Fig. 3. Basis function values are plotted for azimuth [0, 2π]and elevation [0, π2 ]. The above convention to represent HSHbasis function is chosen to imitate the EEG scalp plots.

Utilizing (13) and (22), the hemispherical harmonics de-composition of the signal V (t) can now be expressed as

Vnm(t) =

∫Ω∈S2

V (t,Ω)[Hmn (Ω)]dΩ,

∼=I∑i=1

aiVi(t,Ωi)[Hmn (Ωi)], (24)

where ai is sampling weight [24] and Ωi = (θi, φi) is thelocation of the ith sensor. For finite order n ∈ [0, N ] whereN ≤

√(I)− 1 [25] and m ∈ [−n, n], (24) can be re-written

in a matrix form as

Vnm(t) = HT(Ω)ΓV(t,Ω) (25)

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Proceedings, APSIPA Annual Summit and Conference 2018 12-15 November 2018, Hawaii

010

5

10

HS

H-M

US

IC

105

10

8

Yaxis5 6

X axis

15

42

0 0

X: 2.4Y: 2.4Z: 1.368e+06

X: 5.6Y: 2.8Z: 8.112e+05

010

5

HS

H-M

VD

R

10

10

8

Yaxis

15

5 6X axis4

20 0

X: 2.4Y: 2.4Z: 12.89

X: 5.6Y: 2.8Z: 14.2

Fig. 4: Dipole source localization using HSH-MUSIC and HSH-MVDR for sources placed at (5.6cm, 2.8cm) and(2.4cm, 2.4cm) with 30 dB SNR.

where

Vnm(t) = [V00, V1(−1), V1(0), V1(1), · · · , VNN ]T (26)

Γ = diaga1, a2, a3, . . . aI

(27)

and H(Ω) is a I × (N + 1)2 matrix, whose ith row is definedas

h(Φi) =[H0

0 (Ωi), H−11 (Ωi), H0

1 (Ωi), . . . , HNN (Ωi)

].

(28)

In the presence of additive sensor noise, the spatio-temporaldata model in (11), can be re-written as

[V]I×Ns = [A]I×P [S]P×Ns + [Z]I×Ns (29)

Multiplying both side of equation with HTΓ and utilizing(25), the final hemispherical harmonics domain data model iswritten as

[Vnm](N+1)2×Ns= [Anm](N+1)2×P [S]P×Ns

(30)+[Znm](N+1)2×Ns

,

where Anm = HTΓGM . It is to note that the dimensionalityof data model reduces from I to (N + 1)2, leading tocomputationally efficient approach.

IV. THE DIPOLE SOURCE LOCALIZATION

The hemispherical harmonics domain data model is utilizedherein for source localization. In particular subspace basedMUSIC method and beamforming based MVDR method areproposed. The HSH domain MUSIC expression for dipolesource localization under fixed dipole assumption can now bewritten as

JHSH−MUSIC(r) =1

ATnmP⊥AAnm

(31)

where P⊥A = UI−PUTI−P , UI−P is the set of (I − P ) noise

eigen vectors obtained from eigen value decomposition ofcovarinace matrix R = E[VnmVnm

T ]. Here, G is look-uplead-field vector defined as in (6) and orientation momentsM is assumed to be fixed for fixed dipole assumption. Thelocation of the dipole is estimated from the peak in the HSH-MUSIC spectrum due to orthogonality between signal andnoise subspace.

The HSH MVDR can also be defined in the similar formas

JHSH−MVDR(r) =1

ATnmR−1Anm

(32)

The HSH-MVDR spectrum results in a peak when arraysteering location matches with source location.

Fig. 4 illustrates the HSH-MUSIC and HSH-MDR spectrafor two dipole sources located at (5, 6cm, 2.8cm, 2cm) and(2.4cm, 2.4cm, 2cm). Uniform spatial sampling was takenwith 37 sensors at SNR =30 dB. Two peaks can be observedcorresponding to two sources in both the spectra.

V. PERFORMANCE EVALUATION

Infinite homogeneous isotropic conductor head model withradius of 86 mm was considered for the simulation. Sensorswere placed over hemispherical scalp at the locations shownin Fig. 5. Total 37 electrodes were placed over the scalp. ForHSH MUSIC and HSH MVDR algorithm, inter-grid gap waschosen to be 2 mm. Potential data was generated using forwarddata model considering two point dipole source. Number ofsnapshot was taken to be 100. The sensor noise was taken to beadditive Gaussian in nature. The performance of the proposedmethods has been presented herein using computation time.HSH-MUSIC method is additionally, explored in terms oflocalization accuracy with variation in depth of the dipole.

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Proceedings, APSIPA Annual Summit and Conference 2018 12-15 November 2018, Hawaii

A. Computation TimeFor hemispherical harmonics of order N = 4 the dimen-

sionality of the data model is reduced from 37 in spatialdomain to 25 in HSH domain. The effect is observed incomputation time. System with Intel R©Core

TMi7 processor,

RAM 16 GB, system type 64-bit and clock speed 3.40 GHzwas utilized for the evaluation. Computation time observedfor the four algorithms are shown in Fig. 6. Bar plot withmean and standard deviation of computation time is plotted for50 iterations. HSH-MUSIC method requires less computationtime when compared to conventional MUSIC, conventionalMVDR and HSH-MVDR method.

-6 -4 -2 0 2 4 6

X axis

-6

-4

-2

0

2

4

6

Y a

xis

Fig. 5: Spatial sensor placement over hemispherical scalp.

MUSIC HSH-MUSIC MVDR HSH-MVDR0

0.5

1

1.5

2

2.5

3

3.5

Co

mp

uta

tio

n t

ime

(Sec

)

Fig. 6: Computation time.

B. Localization Accuracy with Depth VariationPerformance of the proposed algorithm is illustrated herein

with variation in depth. The depth of the dipole with co-ordinate (x, y, z)T is computed as

√x2 + y2 + z2. Two

active dipole sources were considered at depth 3.832cmwith co-ordinate (1.3, 3, 2)T , (3.198, 0.676, 2)T . Another setof dipole sources were taken at depth 5cm with co-ordinatesas (3.24, 3.24, 2)T , (4.558, 0.4821, 2)T . The simulation wasperformed considering 30dB SNR. Subspace based methodbeing more accurate, HSH-MUSIC spectrum is plotted herein.Fig. 7 illustrates the effect of depth on the HSH-MUSICspectrum. It is seen that the resolution of the method reducesat the higher depth.

VI. CONCLUSIONS AND FUTURE WORK

Dipole source localization in hemispherical harmonics do-main is addressed for the first time in this paper. Rather thanconventional spherical harmonics, hemispherical harmonicsbasis functions are utilized for DSL. In particular, HSH-MUSIC and HSH-MVDR are proposed for DSL. It is seenthat HSH domain processing is computationally more efficientwhen compared to spatial domain processing. Localization andtracking of dipoles in HSH domain is currently being studied.

0

2

4

6

10

8

10

12

14

HS

H-M

US

IC105

105 Yaxis 5X axis00

X: 1.2Y: 3Z: 6.095e+05

X: 3.2Y: 0.6Z: 1.231e+06

0

1

2

3

10

4

5

6

7

HS

H-M

US

IC

105

105 Yaxis 5X axis00

X: 3.2Y: 3.2Z: 6.296e+05

X: 4.6Y: 1.2Z: 8.162e+04

Fig. 7: HSH MUSIC cost function with depth varied as3.8cm (above) and 5cm (below).

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Proceedings, APSIPA Annual Summit and Conference 2018 12-15 November 2018, Hawaii

ACKNOWLEDGMENT

This work was funded by Prime Minister’s Research Fel-lowship (PMRF), Government of India.

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