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IEEE BIBE 2013

CBP: Cognitive Brain Signal Processing Lab. Towards an Overall 3-D Vector Field Reconstruction via Discretization and a Linear Equations System. Chrysa Papadaniil, Student Member Leontios Hadjileontiadis , Senior Member Aristotle University of Thessaloniki. IEEE BIBE 2013

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IEEE BIBE 2013

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  1. CBP: Cognitive Brain Signal Processing Lab Towards an Overall 3-D Vector Field Reconstruction via Discretization and a Linear Equations System Chrysa Papadaniil, Student Member Leontios Hadjileontiadis, Senior Member Aristotle University of Thessaloniki IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  2. CBP: Cognitive Brain Signal Processing Lab Motivation - EEG based Source Localization ? ? ? ? Forward Problem Inverse Problem Ill posed IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  3. CBP: Cognitive Brain Signal Processing Lab 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 IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  4. CBP: Cognitive Brain Signal Processing Lab 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) IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  5. CBP: Cognitive Brain Signal Processing Lab 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 IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  6. CBP: Cognitive Brain Signal Processing Lab VFT – Literature • 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 • 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 IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  7. CBP: Cognitive Brain Signal Processing Lab 2D-VFT Formulation • Bounded square domain, • grid B • Recovery of the field in the centers of the tiles Δs Q • Ideal point sensors regularly placed at the domain ‘s border • Tracing line connecting two boundary sensors A • Starting from the foot of perpendicular, we discretize the line with a step of Δs P IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  8. CBP: Cognitive Brain Signal Processing Lab 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 IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  9. CBP: Cognitive Brain Signal Processing Lab 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 IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  10. CBP: Cognitive Brain Signal Processing Lab 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 IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  11. CBP: Cognitive Brain Signal Processing Lab 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: • 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 • Starting from A, we sample the line • Sampling points coordinates • increase by: , , • Number of points on the segments: • Coordinates of all sampling points: , , , IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  12. CBP: Cognitive Brain Signal Processing Lab 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 IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  13. CBP: Cognitive Brain Signal Processing Lab 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 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  14. Some Results • 1 point source at (11, 11, 11) • 648 unknowns • 19440 equations IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  15. Some Results • 4 point sources at (10, 10, 10), • (-10, 10, -10), (10, 10, 0), • (-10, -10, 0) IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  16. CBP: Cognitive Brain Signal Processing Lab Conclusion – Future work • Next Steps • Sampling bounds study for the 3D space • Advanced techniques of discretizing the 3D field domain (FEM) • More realistic head models • 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. IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  17. CBP: Cognitive Brain Signal Processing Lab ++pic from cbp.iti • Goals • Advancing the state of the art in vector field tomography • Brain cognitive processes research IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  18. CBP: Cognitive Brain Signal Processing Lab • EGI 300 geodesic system • High resolution data acquisition (dEEG) • 256 channels • Full head coverage • Patient friendly *Pictures from www.egi.com IEEE BIBE 2013 13th IEEE International Conference on BioInformatics and BioEngineering, November 10-13, Chania, Greece

  19. Thank you!

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