ieee bibe 2013 13th ieee international conference on bioinformatics and bioengineering, november...

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



Motivation - EEG based Source Localization

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Forward Problem

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Inverse ProblemIll posed

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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



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)



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



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



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



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



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



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



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



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



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


Some Results

• 1 point source at (11, 11, 11)

• 648 unknowns

• 19440 equations


Some Results

• 4 point sources at (10, 10, 10),

(-10, 10, -10), (10, 10, 0),

(-10, -10, 0)



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


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


EGI 300 geodesic system

High resolution data acquisition (dEEG)

256 channels

Full head coverage

Patient friendly


*Pictures from www.egi.com

Thank you!

ieee bibe 2013 13th ieee international conference on bioinformatics and bioengineering, november...

Documents

greece slide

greece representation

active brain areas

illposed problem slide

greece vft formula

d field

greece p recovery

d vector field reconstruction