Spatial Harmonic Analysis of EEG Data -...
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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
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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
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 4 / 21
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]
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 5 / 21
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]
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 5 / 21
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
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 6 / 21
Material and methodsEigenspaces of the Discrete Laplace-Beltrami Operator
• 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
Material and methodsEigenspaces of the Discrete Laplace-Beltrami Operator
• 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
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 8 / 21
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
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 9 / 21
Material and methodsSpatial decomposition using spatial harmonic basis functions
B,
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 10 / 21
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
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 11 / 21
ApplicationsAnalysis of SEP data
• Evoked activities• parietal, P14 and N20• frontal, P14, P20 and N30
• Spatial harmonic decompositionof the SEP data
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Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 12 / 21
ApplicationsAnalysis of SEP data
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 13 / 21
ApplicationsAnalysis of SEP data
• Contribution of the spatial harmonic basis functionsto the global field power
• Seven basis functions are sufficient to describe90% of the signal energy
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Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 14 / 21
ApplicationsGroup analysis of SEP data
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P14
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P20�N20
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P14
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N30
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 15 / 21
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
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 16 / 21
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
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 17 / 21
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)
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 18 / 21
AcknowledgmentsInstitute of Biomedical Engineering and InformaticsIlmenau University of TechnologyRoland EichardtPatrique FiedlerJens HaueisenDaniel Strohmeier
eemagine Medical Solutions GmbHRalph HauffeJacob KanevFrank Zanow
Grant No. KF2250111ED2
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 19 / 21
Thank you for your attention!
Uwe Graichen (TU Ilmenau) Spatial Harmonic Analysis of EEG Data 2012 20 / 21
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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