# Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Model

#### 2.2. Beamforming and Neural Indices

#### 2.3. Generalized Sidelobe Canceler

#### 2.4. Adaptive Blocking Matrix

#### 2.5. Proposed Reduced-Rank Beamforming Scheme

#### 2.6. EEG Data

#### 2.6.1. Simulated Data

- signal-to-measurement-noise ratio, given by$${\eta}_{m}=20log\frac{{\Vert A\left(\mathit{\theta}\right)Q\Vert}_{\mathrm{F}}}{{\Vert {V}_{k}\Vert}_{\mathrm{F}}},$$
- signal-to-biological-noise ratio, given by$${\eta}_{b}=20log\frac{{\Vert A\left(\mathit{\theta}\right)Q\Vert}_{\mathrm{F}}}{{\Vert A\left({\mathit{\theta}}_{\Delta}\right){Q}_{\Delta}\Vert}_{\mathrm{F}}}.$$

#### 2.6.2. Real EEG Data

## 3. Results

#### 3.1. Evaluation of Performance under Different ${\eta}_{b}$ and ${\eta}_{m}$ Conditions

- the individual bias of the estimates, given by ${b}_{l,k}=\left|\right|{\mathit{r}}_{l}-{\widehat{\mathit{r}}}_{l,k}{\left|\right|}_{2}$;
- their sum-of-squares: ${\mathrm{SS}}_{k}={\sum}_{l=1}^{3}{b}_{l,k}^{2}$;
- the maximum bias: ${b}_{\mathrm{MAX},k}={max}_{l}{b}_{l,k}$.

#### 3.2. Applicability of ${\iota}_{{\mathrm{R}\mathrm{R}}_{1}}$ and ${\iota}_{{\mathrm{R}\mathrm{R}}_{2}}$ in Dipole Source Localization Using Real EEG Data

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Geometrical model with four layers (scalp, skull, brain and thalami). (

**a**) Frontal. (

**b**) Sagittal.

**Figure 3.**Temporal evolution of dipoles ${q}_{1}$ and ${q}_{2}$. Their Cartesian components x, y, and z are shown in red, blue, and green, respectively.

**Figure 4.**Tessellated model of the brain (top view). Red points indicate the $L=3$ real sources’ positions. Other candidate positions that are part of the ROI are shown as blue points.

**Figure 5.**Experimental setup for the acquisition of EEG SSVEP data. Image taken from https://youtu.be/8lGBVvCX5d8 (accessed on 2 February 2023).

**Figure 6.**Results of evaluating $\overline{\mathrm{S}\mathrm{S}}$ for different noise conditions. Bars correspond to $P{\left({W}_{0}\right)}_{\mathrm{min}}$ (yellow), ${\iota}_{\mathrm{MAI}}$ (blue), ${\iota}_{{\mathrm{R}\mathrm{R}}_{1}}$ (red), and ${\iota}_{{\mathrm{R}\mathrm{R}}_{2}}$ (green). (

**a**) Noise conditions: ${\eta}_{b}=10$ dB, ${\eta}_{m}=10$ dB; (

**b**) noise conditions: ${\eta}_{b}=5$ dB, ${\eta}_{m}=5$ dB; (

**c**) noise conditions: ${\eta}_{b}=0$ dB, ${\eta}_{m}=0$ dB; (

**d**) noise conditions: ${\eta}_{b}=-5$ dB, ${\eta}_{m}=0$ dB.

**Figure 7.**Results of evaluating ${\sigma}_{{b}_{\mathrm{max}}}$ for the same noise conditions as in Figure 6. Bars correspond to $P{\left({W}_{0}\right)}_{\mathrm{min}}$ (yellow), ${\iota}_{\mathrm{MAI}}$ (blue), ${\iota}_{{\mathrm{R}\mathrm{R}}_{1}}$ (red), and ${\iota}_{{\mathrm{R}\mathrm{R}}_{2}}$ (green). (

**a**) Noise conditions: ${\eta}_{b}=10$ dB, ${\eta}_{m}=10$ dB; (

**b**) noise conditions: ${\eta}_{b}=5$ dB, ${\eta}_{m}=5$ dB; (

**c**) noise conditions: ${\eta}_{b}=0$ dB, ${\eta}_{m}=0$ dB; (

**d**) noise conditions: ${\eta}_{b}=-5$ dB, ${\eta}_{m}=0$ dB.

**Figure 8.**Results of the source localization process through our reduced-rank beamforming approach in EEG SSVEP data of Subject 5 in [27] and for a specific frequency. Minimum values of ${\iota}_{{\mathrm{R}\mathrm{R}}_{(\xb7)}}$ (shown in darker colors) correspond to zones of main brain activation. (

**a**) ${\iota}_{\mathrm{R}{\mathrm{R}}_{1}}$ (

**b**) ${\iota}_{\mathrm{R}{\mathrm{R}}_{2}}$.

**Figure 9.**Results of the source localization process through our reduced-rank beamforming approach in EEG SSVEP data of Subject 10 in [27] and for a specific frequency. Minimum values of ${\iota}_{{\mathrm{R}\mathrm{R}}_{(\xb7)}}$ (shown in darker colors) correspond to zones of main brain activation. (

**a**) ${\iota}_{\mathrm{R}{\mathrm{R}}_{1}}$ (

**b**) ${\iota}_{\mathrm{R}{\mathrm{R}}_{2}}$.

Variable/Notation | Description |
---|---|

$\mathit{q}$${}_{l}$ | lth dipole source |

L | number of dipoles |

Q | matrix containing all dipole sources |

${q}_{l,x}\left(t\right),{q}_{l,y}\left(t\right),{q}_{l,z}\left(t\right)$ | time-varying magnitudes of lth dipole’s Cartesian components |

N | total number of time samples |

$\mathit{r}$ | vector representing a position (Cartesian coordinates) |

$\Omega $ | volume of the brain |

${\mathit{r}}_{l}$ | lth dipole’s position |

$\mathit{\theta}$ | matrix containing all L dipole’s positions (parameter of interest) |

${y}_{m}\left(t\right)$ | time-varying EEG measurement at the mth sensor |

M | total number of sensors |

${Y}_{k}$ | matrix with all EEG measurements at the kth experiment (trial) |

K | total number of independent trials |

${A}_{l}\left({\mathit{r}}_{l}\right)$ | lth lead field matrix associated to the lth dipole’s position |

$A\left(\mathit{\theta}\right)$ | matrix comprising the L lead field matrices as a function of $\mathit{\theta}$ |

${v}_{m}\left(t\right)$ | measurement noise realization in the mth sensor at time t |

${\sigma}_{V}^{2}$ | variance of measurement noise |

${V}_{k}$ | matrix with all measurement noise at the kth trial |

$\widehat{Z}$ | indicates a consistent estimate of Z |

W | spatial filter (beamformer) |

I | identity matrix |

$\mathbf{0}$ | matrix full of zeros |

$\widehat{Q}$ | indicates an approximation of Q |

${\tilde{Q}}_{k}$ | approximated dipoles at the kth trial |

$A\left(\mathit{r}\right)$ | lead field matrix as a function of $\mathit{r}$ |

$W\left(\mathit{r}\right)$ | beamformer as a function of $\mathit{r}$ |

${W}_{\mathrm{LCMV}}$ | linearly constrained minimum variance (LCMV) beamformer |

R | data covariance matrix |

P | noise covariance matrix |

${\iota}_{\mathrm{LCMV}}\left(\mathit{r}\right)$ | LCMV-based neural activity index (NAI) as a function of $\mathit{r}$ |

${\iota}_{\mathrm{MAI}}\left(\theta \right)$ | multi-source activity index (MAI) as a function of $\mathit{\theta}$ |

$G\left(\theta \right)$ | reciprocal of the noise power as a function of $\mathit{\theta}$ |

$H\left(\theta \right)$ | reciprocal of the sources’ power as a function of $\mathit{\theta}$ |

${W}_{\mathrm{GSC}}$ | generalized sidelobe canceler (GSC) |

${W}_{h}$ | quiescent component of the GSC |

${W}_{0}$ | noise-plus-interference components of the GSC |

${C}_{\perp}$ | blocking matrix |

${P}_{A}^{\perp}$ | projection matrix of A |

${X}_{0}$ | undesired measurement components |

${R}_{{X}_{0}}$ | autocorrelation matrix of the undesired signals |

$\mathbf{b}$ | Wiener filter that minimizes the mean-squares of ${X}_{0}$ |

${Q}_{0}$ | matrix containing all undesired signals |

${\iota}_{\mathrm{R}{\mathrm{R}}_{1}}$ | first proposed reduced-rank (RR) NAI a function of $\mathit{\theta}$ |

${\tilde{W}}_{0}$ | RR approximation of ${W}_{0}$ |

${\lambda}_{j}$ | eigenvalues of ${R}_{{X}_{0}}$ |

${\mathit{u}}_{j}$ | orthonormal eigenvectors of ${R}_{{X}_{0}}$ |

${\iota}_{\mathrm{R}{\mathrm{R}}_{2}}$ | second proposed RR-NAI as a function of $\mathit{\theta}$ |

${R}_{{X}_{0}{Q}_{h}}$ | cross-correlation of ${X}_{0}$ and ${Q}_{h}$ |

${\mathit{\theta}}_{\Delta}$ | matrix with all positions of interference sources |

$A\left({\mathit{\theta}}_{\Delta}\right)$ | lead field matrix as a function of ${\mathit{\theta}}_{\Delta}$ |

${Q}_{\Delta}$ | matrix containing all interference dipole sources |

${\sigma}_{\Delta}^{2}$ | variance of biological noise |

${\eta}_{m}$ | signal-to-measurement-noise ratio |

${\eta}_{b}$ | signal-to-biological-noise ratio |

${\mathit{\theta}}_{\mathrm{cand}}$ | matrix containing candidate dipole’s positions |

${b}_{l,k}$ | bias of the estimate of ${\mathit{r}}_{l}$ at the kth trial |

${\mathrm{SS}}_{k}$ | sum-of-squares of ${b}_{l,k}$ at the kth trial |

${b}_{\mathrm{MAX},k}$ | maximum bias at the kth trial |

$\overline{\mathrm{SS}}$ | average sum-of-squares |

${\sigma}_{{b}_{\mathrm{MAX}}}$ | standard deviation of the maximum bias |

$P{\left({W}_{0}\right)}_{\mathrm{min}}$ | minimum power of ${W}_{0}$ |

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**MDPI and ACS Style**

Jiménez-Cruz, E.; Gutiérrez, D.
Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming. *Appl. Sci.* **2023**, *13*, 3241.
https://doi.org/10.3390/app13053241

**AMA Style**

Jiménez-Cruz E, Gutiérrez D.
Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming. *Applied Sciences*. 2023; 13(5):3241.
https://doi.org/10.3390/app13053241

**Chicago/Turabian Style**

Jiménez-Cruz, Eduardo, and David Gutiérrez.
2023. "Interference Suppression in EEG Dipole Source Localization through Reduced-Rank Beamforming" *Applied Sciences* 13, no. 5: 3241.
https://doi.org/10.3390/app13053241