Tikhonov Example Bio PSE Example Designing and Implementing

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

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

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

Math Derivation Tikhonov Example NCRR

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

Results - Input Tikhonov Example NCRR

Results - Output Tikhonov Example NCRR

Results - Implementation Tikhonov Example NCRR

Pseudocode Tikhonov Example • 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 hgeom • Compute Tikhonov heart voltages ( data) based on hdata, tdata, and R) • Visualize • Plot L-curve, and data vs tdata NCRR

Pseudocode - Input Tikhonov Example • 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 hgeom • Compute Tikhonov heart voltages ( data) based on hdata, tdata, and R) • Visualize • Plot L-curve, and data vs tdata NCRR

Pseudocode – Input Vis Tikhonov Example • 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 hgeom • Compute Tikhonov heart voltages ( data) based on hdata, tdata, and R) • Visualize • Plot L-curve, and data vs tdata NCRR

Pseudocode – Z and R Tikhonov Example • 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 hgeom • Compute Tikhonov heart voltages ( data) based on hdata, tdata, and R) • Visualize • Plot L-curve, and data vs tdata NCRR

Pseudocode – Solve and Vis Tikhonov Example • 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 hgeom • Compute Tikhonov heart voltages ( data) based on hdata, tdata, and R) • Visualize • Plot L-curve, and data vs tdata NCRR

Complete Net Tikhonov Example NCRR