1. Tikhonov Example NCRR BioPSE Example: Designing and Implementing A Tikhonov Regularization Network BioPSE Example: Designing and Implementing A Tikhonov Regularization Network There’s just nothing quite like a live demo…

2. Tikhonov Example NCRR Forward Inverse Forward and Inverse ECG/EEG Forward Problems Bioelectric currents Volume and surface potentials Inverse Problems Identify cardiac sources (ischemia) Localize neural activity (epilepsy, evoked response)

3. Tikhonov Example NCRR Problem Overview Tikhonov regularization Inverse problem validation Given source and measurement surface data Recover source data via regularized inversion Compare to true solution Optimize regularization parameter

4. Tikhonov Example NCRR Math Derivation

5. Tikhonov Example NCRR Problem Specification Input Torso data Heart data Output Plot of residual vs regularized solution (“L-curve”) Visualizations of computed solution vs true solution

6. Tikhonov Example NCRR Results - Input

7. Tikhonov Example NCRR Results - Output

8. Tikhonov Example NCRR Results - Implementation

9. Tikhonov Example NCRR Pseudocode Read in torso nodes with measured data values (t) Read in heart surface with measured data values (h) Visualize t Visualize h Read in transfer matrix Z Compute regularization matrix (R) from h geom Compute Tikhonov heart voltages ( data ) based on h data, t data, and R) Visualize Plot L-curve, and data vs t data

10. Tikhonov Example NCRR Pseudocode - Input Read in torso nodes with measured data values (t) Read in heart surface with measured data values (h) Visualize t Visualize h Read in transfer matrix Z Compute regularization matrix (R) from h geom Compute Tikhonov heart voltages ( data ) based on h data, t data, and R) Visualize Plot L-curve, and data vs t data

11. Tikhonov Example NCRR Pseudocode – Input Vis Read in torso nodes with measured data values (t) Read in heart surface with measured data values (h) Visualize t Visualize h Read in transfer matrix Z Compute regularization matrix (R) from h geom Compute Tikhonov heart voltages ( data ) based on h data, t data, and R) Visualize Plot L-curve, and data vs t data

12. Tikhonov Example NCRR Pseudocode – Z and R Read in torso nodes with measured data values (t) Read in heart surface with measured data values (h) Visualize t Visualize h Read in transfer matrix Z Compute regularization matrix (R) from h geom Compute Tikhonov heart voltages ( data ) based on h data, t data, and R) Visualize Plot L-curve, and data vs t data

13. Tikhonov Example NCRR Pseudocode – Solve and Vis Read in torso nodes with measured data values (t) Read in heart surface with measured data values (h) Visualize t Visualize h Read in transfer matrix Z Compute regularization matrix (R) from h geom Compute Tikhonov heart voltages ( data ) based on h data, t data, and R) Visualize Plot L-curve, and data vs t data

14. Tikhonov Example NCRR Complete Net