Considering New Regularization Parameter-Choice Techniques for the Tikhonov Method to Improve the Accuracy of Electrocardiographic Imaging

Abstract

The electrocardiographic imaging (ECGI) inverse problem highly relies on adding constraints, a process called regularization, as the problem is ill-posed. When there are no prior information provided about the unknown epicardial potentials, the Tikhonov regularization method seems to be the most commonly used technique. In the Tikhonov approach the weight of the constraints is determined by the regularization parameter. However, the regularization parameter is problem and data dependent, meaning that different numerical models or different clinical data may require different regularization parameters. Then, we need to have as many regularization parameter-choice methods as techniques to validate them. In this work, we addressed this issue by showing that the Discrete Picard Condition (DPC) can guide a good regularization parameter choice for the two-norm Tikhonov method. We also studied the feasibility of two techniques: The U-curve method (not yet used in the cardiac field) and a novel automatic method, called ADPC due its basis on the DPC. Both techniques were tested with simulated and experimental data when using the method of fundamental solutions as a numerical model. Their efficacy was compared with the efficacy of two widely used techniques in the literature, the L-curve and the CRESO methods. These solutions showed the feasibility of the new techniques in the cardiac setting, an improvement of the morphology of the reconstructed epicardial potentials, and in most of the cases of their amplitude.

Comparison of parameter optimization methods for quantitative susceptibility mapping

“…The U‐curve method is widely used to find a range of suitable optimal parameters in inverse problems, such as inverse electromagnetic modeling, super‐resolution, and others 17,23,24 . However, it has not been tested yet for QSM problems.…”

“…The U-curve method is widely used to find a range of suitable optimal parameters in inverse problems, such as inverse electromagnetic modeling, super-resolution, and others. 17,23,24 However, it has not been tested yet for QSM problems. This method minimizes a convex functional defined by the sum of the reciprocates of the data consistency and regularization costs (C and R, as defined for the L-curve) 17 :…”

“…This makes the maximum curvature search not robust enough for fine‐tuning, providing only an order of magnitude estimation in most scenarios. In addition, it has been observed that L‐curve–based optimization usually yields overregularized solutions 17 . A different approach using both the data fidelity and reconstruction costs is present in the U‐curve 17 analysis, with promising results.…”

“…In addition, it has been observed that L‐curve–based optimization usually yields overregularized solutions 17 . A different approach using both the data fidelity and reconstruction costs is present in the U‐curve 17 analysis, with promising results. For these reasons, new strategies for optimal parameter fine‐tuning must be developed for QSM applications.…”

In silico validation of electrocardiographic imaging to reconstruct the endocardial and epicardial repolarization pattern using the equivalent dipole layer source model

“…As the L-curve method to find the optimal value for lambda often leads to over-regularization [3], we set beforehand a target value for REG that results in physiologically realistic timing patterns. We found that a REG value of 25 s/m corresponds to realistic activation patterns, while a REG value of 10 s/m corresponds to realistic repolarization patterns.…”