IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
Towards an Overall 3-D Vector Field Reconstruction via Discretization and a Linear Equations System
CBP: Cognitive Brain Signal Processing Lab
Chrysa Papadaniil, Student Member
Leontios Hadjileontiadis, Senior Member
Aristotle University of Thessaloniki
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
Motivation - EEG based Source Localization
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Forward Problem
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Inverse ProblemIll posed
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CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
EEG based Source Localization - Common approach
Representation of the active brain areas using a number of dipoles
Different methodologies:• A priori postulation of the dipoles,
solution of the forward problem, parameters change until the solution agrees with the scalp measurements
• Bayesian estimation
• Beamforming
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
EEG based Source Localization - Alternative approach
Given the measured scalp potentials, what is the electrostatic field inside the head?
Mapping of the brain to a set of active effective states
No a priori assumptions
Reduced complexity (we ignore the electromagnetic properties of different tissues)
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
Vector Field Tomography (VFT)
Methods for the recovery of fields from integral data
• Irrotational, stationary field inside the head from surface measurements ( )
VFT formula for line integrals• Line integral:
• In two dimensions: (Radon Transform)
• In three dimensions: (Ray Transform)
: field to be recovered,
fx, fy, fz: ’s components,
: line direction vector w: angle of L with the positive x-axis, φ,θ: spherical angles
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
VFT – Literature
Recovering a 2D field from integral data is by definition underdetermined – only one component could be determined (irrotational or solenoidal)
Possible solution: Both transversal and longitudinal measurements (Braun and Hauck)
Drawback: Very few applications allow for both kinds measurements
Suggested approach: Instead of working in the continuous domain,
• we reconstruct the field in specific sampling points arranged in a grid, where there is data redundancy
• we may use many line orientations passing through every point and then view their recordings as weighted sums of the local vector field’s Cartesian components
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
2D-VFT Formulation
P
• Bounded square domain, grid
• Recovery of the field in the centers of the tiles
• Ideal point sensors regularly placed at the domain ‘s border
• Tracing line connecting two boundary sensors
𝐬𝑤
A
B
• Starting from the foot of perpendicular, we discretize the line with a step of Δs
Q
Δs
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
2D-VFT Formulation
Each sampling point along the line is assigned to the nearest tile center
We approximate numerically the line integral by i, j represent the tiles enumeration
We use the lines that connect all sensors combinations – the solution stems from the system of the linear equations
Well conditioned system
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
2D-VFT, more work
Improved 2D-VFT reconstruction using probabilistic weights to account for the non uniform placement of the sensors (Radon requirement for medical accuracy image reconstruction)
Sampling bounds for the Radon parameters
Robust formulation • Existence of upper bound to the solution error• Discretization serves as regularization for the ill-posed
problem
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
2D-VFT, more work
Improved 2D-VFT reconstruction using probabilistic weights to account for the non uniform placement of the sensors (Radon requirement for medical accuracy image reconstruction)
Sampling bounds for the Radon parameters
Robust formulation • Existence of upper bound to the solution error• Discretization serves as regularization for the ill-posed
problem
Our first goal: The extension of the methodology to 3 dimensions
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
3D-VFT
• Bounded cubic space, digitized to a grid
• We want to recover on the centers of the tiles
• If we assume the AB segment with boundary points AB ‘s parameters are:
• Unit vector:
• Starting from A, we sample the line
• Sampling points coordinates increase by: , ,
• Number of points on the segments:
• Coordinates of all sampling points: , , ,
• We enumerate the tiles using integers
• Data redundancy achieved by assigning the sampling points to the closest tile center by:
• Numerical approximation:
• Ideal sensors placed in the centers of the outward faces of the boundary tiles
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
3D-VFT We follow the same procedure for all the combinations of
sensors apart from the ones lying in the same face
The unknown field components are
The resulting equations are
The system of equations can be synopsized as: () contains the sensors measurements () contains the unknowns () is the system matrix with the coefficients connecting each scanning line with the corresponding field values
, well conditioned system
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
Simulations setup
We considered fields produced by electric monopoles (irrotational)• Estimation of the right part of the integrals by the voltage difference between two sensors points• Simulation of the field inside the head
The theoretical field and the voltage values in the sensors locations were calculated using Coulomb’s law
b was determined from all the sensors combinations differences and A using the methodology presented
Relative and angular errors estimated for comparison
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
Some Results
• 1 point source at (11, 11, 11)
• 648 unknowns
• 19440 equations
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
Some Results
• 4 point sources at (10, 10, 10),
(-10, 10, -10), (10, 10, 0),
(-10, -10, 0)
CBP: Cognitive Brain Signal Processing Lab
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
Conclusion – Future work
The discretization of both the field domain and the
scanning lines creates data redundancy, allowing for
the recovery of all the components of the unknown
3D field only from boundary data.
Next Steps
• Sampling bounds study for the 3D space
• Advanced techniques of discretizing the 3D field domain (FEM)
• More realistic head models
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
CBP: Cognitive Brain Signal
Processing Lab
Goals• Advancing the state of the art in
vector field tomography
• Brain cognitive processes research
++pic from cbp.iti
IEEE BIBE 201313th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece
EGI 300 geodesic system
High resolution data acquisition (dEEG)
256 channels
Full head coverage
Patient friendly
CBP: Cognitive Brain Signal Processing Lab
*Pictures from www.egi.com
Thank you!