Spatial Harmonic Analysis of EEG Data -...

Spatial Harmonic Analysis of EEG Data

Uwe Graichen

Institute of Biomedical Engineering and InformaticsIlmenau University of Technology

Singapore, 8/11/2012

Outline

1 Motivation

2 Introduction

3 Material and methods

4 Applications

5 Summary

Motivation• EEG – important diagnostic tool to investigate the brain function• Up to 512 recording channels at sample rates of up to 20 kHz• Considerable quantity of data, particularly for long term

measurements• Efficient signal analysis and decomposition methods are essential• Investigation of spatial distribution of multichannel EEG is of

particular interest

-100 0 100 200 300 400

-5

0

5

t@msD

V@m

VD

Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 3 / 21

IntroductionSpatial decomposition of multichannel EEG dataState of the art

• Principal Component Analysis (PCA) [Lagerlund et al., 1997]• Independent Component Analysis (ICA) [Jung et al., 2001]• Parallel Factor Analysis (PARAFAC) [Miwakeichi et al., 2004]• Matching Pursuit [Gratkowski et al., 2008, 2007]

Proposed Approach• New method for spatial harmonic analysis of EEG data using the

Laplacian eigenspace of the meshed surface of electrode positions• Generation of spatial harmonics basis functions for arbitrary

arrangements of EEG electrodes• Fast generation of basis functions and fast decomposition of data• In addition this approach facilitates the rejection of noisy and

erroneous components, the Improvement of source localization andthe compression of data


Material and methodsEigenspaces of the Continuous Laplace-Beltrami Operator

• Laplace-Beltrami operator ∆ for a function f ∈ C2 on a manifold Mis defined by

∆f = div(grad f )

• Solving the Laplacian eigenvalue problem

∆φ = λφ

with λ := −k2 −→ ∆φ+ k2φ = 0 (Helmholtz equation)

• Eigenfunctions φ form a set of basis functions for a harmonicanalysis on a manifold

• The Laplacian eigenspace can be considered as a basis of ageneralized Fourier analysis [Chavel, 1984; Rosenberg, 1997;Berger, 2003]


Material and methodsEigenspaces of the Discrete Laplace-Beltrami Operator

• Surfaces are often represented by triangulated meshes M = {V ,E ,F}• Discretization of the Laplace-Beltrami operator ∆ for a function

f : V → R using FEM approach• Matrix notation of the Laplace-Beltrami operator ∆~f = −L~f• Laplacian matrix L = B−1Q

with mass matrix B and stiffness matrix Q

tb

ta

v

vi

jαij

eij

βij

vk

vl



• Mass matrix Bij =

(∑

t∈iO |t|) /6 if i = j(|ta|+ |tb|) /12 if eij ∈ E0 otherwise

• Stiffness matrix Qij =

∑

j Qij if i = j−1

2 (cot(αij) + cot(βij)) if eij ∈ E0 otherwise

tb

ta

v

vi

jαij

eij

βij

vk

vlUwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 7 / 21


• Computation of the basis functions using a generalized symmetricdefinite eigenproblem (FEM approach)

− Q~x = λB~x

• Modification of the inner product using the mass matrix B, to assurethe B-orthogonality ⟨

~f , ~x⟩

B= ~f >B ~x

• Normalization by dividing each eigenvector ~xi by its B-relative norm

‖~xi‖B =√〈~xi , ~xi〉B

• Eigenvectors ~x form a harmonic orthonormal basis and can be usedfor a spectral analysis of functions defined on the mesh M


Material and methodsGeneration of harmonic spatial basis functions

• Determination of the mass matrix B and stiffness matrix Q using thetopology of EEG setup and electrode positions

• Computation of the basis functions by solving the generalizedsymmetric definite eigenproblem (FEM approach)

−Q~x = λB~x

• Resulting eigenvectors ~x form a set of spatial harmonic basis functions


Material and methodsSpatial decomposition using spatial harmonic basis functions

B,


Material and methodsData

• Somatosensory-evoked potentials (SEP), transcutaneous electricalstimulation of nervus medianus

• 11 Subjects, 6000 stimulations, stimulus frequency 3.7Hz• 256-channel EEG head-cap, equidistant electrode layout, sampling

frequency 2048Hz• Electrode positions tracked by an optical 3D electrode digitizer system

R6A

R12R

R11RR10R

R12Z

R11Z

R5HR4H

R7GR6G R5G

R6FR7F

R9ER8E

R8D

R7E

R7D R6D

R7CR6C

R6BR5B

R5A

R9R R8R

R10ZR8ZR9Z

R5F R4F

R5ER6E

R5D R4D

R5C R4C

R3BR4B

R3AR4A

R1A

R2A

R7RR6R

R5R

R6Z

R7Z

R3H

R1HR2H

R4G

R2GR3G

R2FR3F

R2ER3ER4E

R2DR3D

R2CR3C

R1BR2B

R1R R1Z

Z1RR2ZR2R

Z2R

R3R

R4R

R3Z

Z3RR4Z

Z4RR5Z

R1G

R1F

R1E

R1D

R1C


ApplicationsAnalysis of SEP data

• Evoked activities• parietal, P14 and N20• frontal, P14, P20 and N30

• Spatial harmonic decompositionof the SEP data

-50 0 50 100 150

-1

0

1

2

3

V�µ

V



• Contribution of the spatial harmonic basis functionsto the global field power

• Seven basis functions are sufficient to describe90% of the signal energy

0 20 40 60 80 1000

50

100

150

200

t@msD

P@m

V2

D


ApplicationsGroup analysis of SEP data

0 10 20 30 40 50 60 70 80 90 100% Pges

2 4 6 8 1015

10152025

No. Sub.

No

.BF

P14

2 4 6 8 1015

10152025

No. Sub.

No

.BF

P20�N20

2 4 6 8 1015

10152025

No. Sub.

No

.BF

N30

è

èè

è

è

èè

è

è

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

20

40

60

80

No. BF

%P

ges

P14

èè

è

è

èè

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

20

40

60

80

No. BF

%P

ges

P20�N20

è

è

è

è

è

èè

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

20

40

60

80

No. BF

%P

ges

N30


Further applicationsArtifact detection

• Training and test set, 1000 samples with eye blink artifacts and 1000samples without eye blink artifacts, randomly chosen

• 1000 repetitions, standardizing (zero mean, unit sample variance)• Fisher discriminant analysis on spatial harmonic decomposed data

• correct classification: 97.8% eye blink and 91.3% non eye blinksamples


Further applicationsData compression

• 128 channel VEP data, 105ms, 118ms and 190ms after stimulation• Data reconstruction using only 3, 10 or 20 low frequency BF• compression ratio (CR) 2.34%, 7.81% and 15.62%

orig 2.34% 7.81% 15.62%

105ms

118ms

190ms


Summary

• New spatial harmonic analysis method for EEG data usingeigenvectors of the discrete Laplace-Beltrami operator

• Adaptive harmonic orthonormal basis, computed using the topologyof the EEG montage and the electrode positions in R3

• Application to arbitrary electrode setups and furthermore to othersensor arrays like MEG

• Wide range of potential applications for non-regular sensor setups(also outside the field of biomedical engineering)


AcknowledgmentsInstitute of Biomedical Engineering and InformaticsIlmenau University of TechnologyRoland EichardtPatrique FiedlerJens HaueisenDaniel Strohmeier

eemagine Medical Solutions GmbHRalph HauffeJacob KanevFrank Zanow

Grant No. KF2250111ED2


Thank you for your attention!


Bibliography I

M. Berger. A Panoramic View of Riemannian Geometry. Springer, 2003. ISBN 978-3540653172.

I. Chavel. Eigenvalues in Riemannian Geometry, volume 115 of Pure and Applied Mathematics. Academic Press, 1984. ISBN978-0121706401.

M. Gratkowski, J. Haueisen, L. Arendt-Nielsen, and F. Zanow. Topographic matching pursuit of spatio-temporalbioelectromagnetic data. Przeglad Elektrotechniczny, 83(11):138–141, 2007.

M. Gratkowski, J. Haueisen, L. Arendt-Nielsen, A. C. N. Chen, and F. Zanow. Decomposition of biomedical signals in spatial andtime-frequency modes. Methods of Information in Medicine, 47(1):26–37, 2008. ISSN 0026-1270. doi: 10.3414/ME0355.

T. P. Jung, S. Makeig, M. Westerfield, J. Townsend, E. Courchesne, and T. J. Sejnowski. Analysis and visualization ofsingle-trial event-related potentials. Human Brain Mapping, 14(3):166–185, 2001. ISSN 1065-9471.

T. D. Lagerlund, F. W. Sharbrough, and N. E. Busacker. Spatial filtering of multichannel electroencephalographic recordingsthrough principal component analysis by singular value decomposition. Journal of Clinical Neurophysiology, 14(1):73–82,1997. ISSN 0736-0258.

F. Miwakeichi, E. Martinez-Montes, P. A. Valdes-Sosa, N. Nishiyama, H. Mizuhara, and Y. Yamaguchia. Decomposing EEG datainto space-time-frequency components using parallel factor analysis. Neuroimage, 22(3):1035–1045, 2004. ISSN 1053-8119.

S. Rosenberg. The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds. Number 31 in LondonMathematical Society Student Texts. Cambridge University Press, 1997.


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