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SPHARA--a generalized spatial Fourier analysis for multi-sensor systems with non-uniformly arranged sensors: application to EEG.

Graichen U, Eichardt R, Fiedler P, Strohmeier D, Zanow F, Haueisen J - PLoS ONE (2015)

Bottom Line: Important requirements for the analysis of multichannel EEG data are efficient techniques for signal enhancement, signal decomposition, feature extraction, and dimensionality reduction.Using FEM, to recover 95% and 99% of the total energy of the EEG data, on average only 35% and 58% of the coefficients are necessary.We conclude that SPHARA can be used for spatial harmonic analysis of multi-sensor data at arbitrary positions and can be utilized in a variety of other applications.

View Article: PubMed Central - PubMed

Affiliation: Institute of Biomedical Engineering and Informatics, Faculty of Computer Science and Automation, Technische Universität Ilmenau, Ilmenau, Germany.

ABSTRACT
Important requirements for the analysis of multichannel EEG data are efficient techniques for signal enhancement, signal decomposition, feature extraction, and dimensionality reduction. We propose a new approach for spatial harmonic analysis (SPHARA) that extends the classical spatial Fourier analysis to EEG sensors positioned non-uniformly on the surface of the head. The proposed method is based on the eigenanalysis of the discrete Laplace-Beltrami operator defined on a triangular mesh. We present several ways to discretize the continuous Laplace-Beltrami operator and compare the properties of the resulting basis functions computed using these discretization methods. We apply SPHARA to somatosensory evoked potential data from eleven volunteers and demonstrate the ability of the method for spatial data decomposition, dimensionality reduction and noise suppression. When employing SPHARA for dimensionality reduction, a significantly more compact representation can be achieved using the FEM approach, compared to the other discretization methods. Using FEM, to recover 95% and 99% of the total energy of the EEG data, on average only 35% and 58% of the coefficients are necessary. The capability of SPHARA for noise suppression is shown using artificial data. We conclude that SPHARA can be used for spatial harmonic analysis of multi-sensor data at arbitrary positions and can be utilized in a variety of other applications.

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The approximation of the Laplace-Beltrami operator using Eq (9).(a) continuous representation; (b) discrete representation. The neighborhood i⋆ of vertex vi consists of the vertices {vx ∈ V : eix ∈ E}. Either the length of eij or the size of the two angles αij and βij opposed to the edge eij are used to estimate the weight w(i, j) for eij. The two triangles ta and tb both share the edge eij; (c) the area of the barycell  for the vertex vi.
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pone.0121741.g001: The approximation of the Laplace-Beltrami operator using Eq (9).(a) continuous representation; (b) discrete representation. The neighborhood i⋆ of vertex vi consists of the vertices {vx ∈ V : eix ∈ E}. Either the length of eij or the size of the two angles αij and βij opposed to the edge eij are used to estimate the weight w(i, j) for eij. The two triangles ta and tb both share the edge eij; (c) the area of the barycell for the vertex vi.

Mentions: For any point pi on 𝓜, an approximation ΔA for Δ is given by the curvilinear integralΔAf(pi)=1/γ/∫p∈γ(f(pi)-f(p))dp,(9)where γ is a closed simple curve on 𝓜 surrounding the point pi, and ∣γ∣ is the length of γ [40, 41], see Fig 1(a). In practical applications, only a few discrete points of γ are usually known. In the recording of EEG data, the continuous function f describing the potential distribution on the surface of the head is sampled only at the electrode positions. For this reason, discretization approaches for Δ are necessary. If a geometric discretization approach is used, the discrete Laplace-Beltrami operator converges to the continuous form ΔD → Δ when the grid is refined [38].


SPHARA--a generalized spatial Fourier analysis for multi-sensor systems with non-uniformly arranged sensors: application to EEG.

Graichen U, Eichardt R, Fiedler P, Strohmeier D, Zanow F, Haueisen J - PLoS ONE (2015)

The approximation of the Laplace-Beltrami operator using Eq (9).(a) continuous representation; (b) discrete representation. The neighborhood i⋆ of vertex vi consists of the vertices {vx ∈ V : eix ∈ E}. Either the length of eij or the size of the two angles αij and βij opposed to the edge eij are used to estimate the weight w(i, j) for eij. The two triangles ta and tb both share the edge eij; (c) the area of the barycell  for the vertex vi.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4401437&req=5

pone.0121741.g001: The approximation of the Laplace-Beltrami operator using Eq (9).(a) continuous representation; (b) discrete representation. The neighborhood i⋆ of vertex vi consists of the vertices {vx ∈ V : eix ∈ E}. Either the length of eij or the size of the two angles αij and βij opposed to the edge eij are used to estimate the weight w(i, j) for eij. The two triangles ta and tb both share the edge eij; (c) the area of the barycell for the vertex vi.
Mentions: For any point pi on 𝓜, an approximation ΔA for Δ is given by the curvilinear integralΔAf(pi)=1/γ/∫p∈γ(f(pi)-f(p))dp,(9)where γ is a closed simple curve on 𝓜 surrounding the point pi, and ∣γ∣ is the length of γ [40, 41], see Fig 1(a). In practical applications, only a few discrete points of γ are usually known. In the recording of EEG data, the continuous function f describing the potential distribution on the surface of the head is sampled only at the electrode positions. For this reason, discretization approaches for Δ are necessary. If a geometric discretization approach is used, the discrete Laplace-Beltrami operator converges to the continuous form ΔD → Δ when the grid is refined [38].

Bottom Line: Important requirements for the analysis of multichannel EEG data are efficient techniques for signal enhancement, signal decomposition, feature extraction, and dimensionality reduction.Using FEM, to recover 95% and 99% of the total energy of the EEG data, on average only 35% and 58% of the coefficients are necessary.We conclude that SPHARA can be used for spatial harmonic analysis of multi-sensor data at arbitrary positions and can be utilized in a variety of other applications.

View Article: PubMed Central - PubMed

Affiliation: Institute of Biomedical Engineering and Informatics, Faculty of Computer Science and Automation, Technische Universität Ilmenau, Ilmenau, Germany.

ABSTRACT
Important requirements for the analysis of multichannel EEG data are efficient techniques for signal enhancement, signal decomposition, feature extraction, and dimensionality reduction. We propose a new approach for spatial harmonic analysis (SPHARA) that extends the classical spatial Fourier analysis to EEG sensors positioned non-uniformly on the surface of the head. The proposed method is based on the eigenanalysis of the discrete Laplace-Beltrami operator defined on a triangular mesh. We present several ways to discretize the continuous Laplace-Beltrami operator and compare the properties of the resulting basis functions computed using these discretization methods. We apply SPHARA to somatosensory evoked potential data from eleven volunteers and demonstrate the ability of the method for spatial data decomposition, dimensionality reduction and noise suppression. When employing SPHARA for dimensionality reduction, a significantly more compact representation can be achieved using the FEM approach, compared to the other discretization methods. Using FEM, to recover 95% and 99% of the total energy of the EEG data, on average only 35% and 58% of the coefficients are necessary. The capability of SPHARA for noise suppression is shown using artificial data. We conclude that SPHARA can be used for spatial harmonic analysis of multi-sensor data at arbitrary positions and can be utilized in a variety of other applications.

Show MeSH
Related in: MedlinePlus