EEG Source Localization: Inverse Problem Solving via Regularization & Truncated SVD

Academic computational project developed for the Numerical Methods for Data Mining course (Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli Federico II).

This project models the electroencephalography (EEG) forward problem within a 3D spherical volume conductor ($R = 20\text{ cm}$) and compares numerical regularization techniques to solve the underdetermined inverse problem of neural source localization.

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📌 Executive Summary

Electroencephalography (EEG) source localization aims to estimate intracranial neural current dipole activity from non-invasive scalp electric potentials.

The overall framework consists of two complementary stages:

• The Forward Problem: Computes the electric potentials recorded at the surface electrodes given an assumed configuration of intracranial current sources within a discretized volume conductor model.
• The Inverse Problem: Reconstructs the unknown spatio-temporal source activation profiles from the observed multichannel electrode recordings.

In this setup, the head volume conductor is discretized as a homogenous sphere of radius $R = 20\text{ cm}$ containing $N = 32$ distributed neural sources and $M = 5$ surface electrodes over $T = 100$ discrete sampling instances.

The linear mapping relating internal dipole activities to scalp recordings is governed by:

$$V = G x + \epsilon, \quad M \ll N$$

where:

• $V \in \mathbb{R}^{5 \times 100}$ represents the measured potential matrix across all channels over time.
• $x \in \mathbb{R}^{32 \times 100}$ denotes the unknown source intensity matrix.
• $G \in \mathbb{R}^{5 \times 32}$ is the Lead Field Matrix, encoding head geometry and conductive properties via an inverse-squared Euclidean distance metric.
• $\epsilon \in \mathbb{R}^{5 \times 100}$ denotes additive measurement noise.

Because the number of active dipole sources significantly exceeds the number of recording sensors ($M \ll N$), the inverse system is severely underdetermined and ill-conditioned, possessing an infinite number of admissible solutions.

This repository demonstrates:

1. Geometric Forward Modeling: Construction of a spherical conductor ($R = 20\text{ cm}$), interior dipole coordinates ($N = 32$), and surface electrode placement ($M = 5$).
2. Dynamic Signal Simulation: Multichannel coupled autoregressive (AR) dynamics over 100 time points with additive Gaussian noise.
3. Analytic Minimum-Norm Solution: Minimum $L^2$-norm regularized reconstruction using the right Moore-Penrose pseudo-inverse $x = G^T (G G^T)^{-1} V$.
4. Truncated SVD Thresholding: Moore-Penrose pseudo-inversion across singular value cutoff tolerances $\tau \in {10^{-1}, 10^{-2}, 10^{-3}, 10^{-4}}$ to evaluate reconstruction fidelity and numerical stability.

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🧠 Mathematical Formulation

1. Forward Model & Lead Field Matrix ($G$)

Derived from electrostatic point-source field decay:

$$E(r, t) = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^N \frac{q_i(t) \mathbf{a}_i(t)}{R^2}$$

where $R = \Vert r - r_i \Vert$, $q_i(t)$ is the signal from source $i$, and $\mathbf{a}_i(t)$ is a unit vector pointing in the direction of the line between the charge and the field point $r$.

The lead field matrix $G$ is constructed using the following approximation equation, where the coupling coefficient between electrode channel $i$ and source dipole $j$ is modeled as:

$$G_{ij} = \frac{c}{R^2}, \quad c = 1$$

2. Inverse Problem & Regularization Framework

The inverse problem consists of estimating the source activity values that generated the measured electric potential field vector at the electrodes. The general strategy formulates this estimation as a regularized linear optimization problem:

$$\hat{x} = \min_x \left( \Vert V - Gx \Vert_2^2 + \sum_{i=1}^k \alpha_i \Vert W_i x\Vert_p \right)$$

where:

• $k$ is the number of regularization constraints reflecting a priori physiological information.
• $W_i \in \mathbb{R}^{5 \times 32}$ are weighting matrices associated with the imposed constraints.
• $\alpha_i > 0$ are the regularization parameters controlling the trade-off and relative importance of each penalty term.

3. Inverse Methods Comparison

• Minimum $L^2$-Norm Regularization: Setting $k=1$, $p=2$, and identity weighting simplifies the unconstrained optimization into the minimum $L^2$-norm solution via the right pseudo-inverse: $$x_{\text{reg}} = G^T (G G^T)^{-1} V$$
• Moore-Penrose Pseudo-Inverse (SVD-based): Computed via Singular Value Decomposition with singular-value thresholding tolerance $\tau$: $$x_{\text{pinv}} = G^\dagger_\tau V$$

4. Key Findings

• A tolerance threshold of $\tau = 10^{-4}$ guarantees numerical robustness, matching the analytic minimum-norm reconstruction with machine-precision residual error ($|V - G x| \sim 10^{-15}$).
• A tolerance of $\tau = 10^{-3}$ excessively truncates informative singular components, resulting in non-negligible reconstruction distortion ($\sim 10^0$).

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📊 Visualizations & Diagnostic Outputs

• 3D Geometry: Visual representation of the 32 internal sources inside the spherical volume conductor alongside the 5 surface electrodes.
• Electrode Tracking: Multi-panel comparisons across all 5 channels showing ground truth $V_i(t)$, regularized reconstruction, and pseudo-inverses with $\tau = 10^{-4}$ and $\tau = 10^{-3}$.
• Residual Contour Maps: Space-time contour plots ($5\text{ channels} \times 100\text{ samples}$) of absolute tracking error $|V - G \hat{x}|$.

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🚀 Getting Started

Prerequisites

• MATLAB (R2019b or newer recommended; requires tiledlayout).
• Runs entirely on core MATLAB linear algebra routines (no additional toolboxes required).

Running the Code

1. Clone this repository:

   git clone [https://github.com/chiaragiavalisco/eeg-source-localization-regularization.git](https://github.com/chiaragiavalisco/eeg-source-localization-regularization.git)
   cd eeg-source-localization-regularization
   

2. Open MATLAB, navigate to the cloned folder, and run:

   run('eeg_source_localization.m')

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🛠 Skills & Competencies Demonstrated

• Numerical Linear Algebra: Underdetermined linear systems, Moore-Penrose pseudo-inverses, Singular Value Decomposition (SVD), rank conditioning, and spectral matrix norms.
• Regularization Theory: Tikhonov / Minimum-norm formulation, Truncated SVD, noise sensitivity, and hyperparameter/threshold tuning.
• Biomedical Modeling: Biophysical volume conduction, forward/inverse EEG problem modeling, lead field matrix formulation.
• MATLAB Computing: Vectorized computation, 3D geometric visualization, dynamic simulation, and publication-ready multi-axis plotting.

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👤 Author

Chiara Giavalisco

• Master's Degree Coursework: Numerical Methods for Data Mining
• LinkedIn Profile • GitHub Profile • Email

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📚 References

1. Jatoi, M. A., Kamel, N., Malik, A. S., Faye, I., & Begum, T. (2014). A survey of methods used for source localization using EEG signals. Biomedical Signal Processing and Control, 11, 42–52.
2. Galaris, E., Gallos, I., Myatchin, I., Lagae, L., & Siettos, C. (2020). Electroencephalography source localization analysis in epileptic children during a visual working-memory task. International Journal for Numerical Methods in Biomedical Engineering, 36(5), e3404.
3. Mégevand, P., & Seeck, M. (2018). Electroencephalography, magnetoencephalography and source localization: their value in epilepsy. Current Opinion in Neurology, 31(2), 176–183.

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📄 License

This project is open-source and available under the MIT License.