eeg-source-localization-regularization
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.
📌 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:
- Geometric Forward Modeling: Construction of a spherical conductor ($R = 20\text{ cm}$), interior dipole coordinates ($N = 32$), and surface electrode placement ($M = 5$).
- Dynamic Signal Simulation: Multichannel coupled autoregressive (AR) dynamics over 100 time points with additive Gaussian noise.
- Analytic Minimum-Norm Solution: Minimum $L^2$-norm regularized reconstruction using the right Moore-Penrose pseudo-inverse $x = G^T (G G^T)^{-1} V$.
- 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.
🧠 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 ($\sim 10^0$).
📊 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} $.
🚀 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
-
Clone this repository: ```bash git clone https://github.com/chiaragiavalisco/eeg-source-localization-regularization.git cd eeg-source-localization-regularization
-
Open MATLAB, navigate to the cloned folder, and run:
run('eeg_source_localization.m')
🛠 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.
👤 Author
Chiara Giavalisco
- Master’s Degree Coursework: Numerical Methods for Data Mining
- LinkedIn Profile • GitHub Profile • Email
📚 References
- 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.
- 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.
- Mégevand, P., & Seeck, M. (2018). Electroencephalography, magnetoencephalography and source localization: their value in epilepsy. Current Opinion in Neurology, 31(2), 176–183.
📄 License
This project is open-source and available under the MIT License.