Improving the Spatial Solution of Electrocardiographic Imaging: A New Regularization Parameter Choice Technique for the Tikhonov Method

Abstract

International audienceThe electrocardiographic imaging (ECGI) inverse problem is highly ill-posed and regularization is needed to stabilize the problem and to provide a unique solution. When Tikhonov regularization is used, choosing the regulariza-tion parameter is a challenging problem. Mathematically, a suitable value for this parameter needs to fulfill the Discrete Picard Condition (DPC). In this study, we propose two new methods to choose the regularization parameter for ECGI with the Tikhonov method: i) a new automatic technique based on the DPC, which we named ADPC, and ii) the U-curve method, introduced in other fields for cases where the well-known L-curve method fails or provides an over-regularized solution , and not tested yet in ECGI. We calculated the Tikhonov solution with the ADPC and U-curve parameters for in-silico data, and we compared them with the solution obtained with other automatic regularization choice methods widely used for the ECGI problem (CRESO and L-curve). ADPC provided a better correlation coefficient of the potentials in time and of the activation time (AT) maps, while less error was present in most of the cases compared to the other methods. Furthermore, we found that for in-silico spiral wave data, the L-curve method over-regularized the solution and the AT maps could not be solved for some of these cases. U-curve and ADPC provided the best solutions in these last cases

An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

“…(33). Anyhow, it seems that careful modifications of well known strategies can be applied under the 2 nd assumption of a priori information, strategies like discrepancy principle, L−curve [15], U -curve, balancing principle [23], or ADP technique [12].…”

“…, a quasi solution regularization scheme is immediately suggested from (14) to the inverse problem (12) when D (A) = L 2 (Γ 1 ). As it will be shown shortly, the admissible data solution (AD solution) is indeed a quasi solution scheme with a slightly more regular definition of (14) as cornerstone.…”

“…The characterization of the admissible data set immediately follows from (12) and the analytic form of mappings A and B.…”

“…The boundary data completion problem in Γ a has been approached as the first kind linear inverse problem in eq. (12). That can also be done in the framework of the invariant embedding in [2], with the slightly difference that the embedding is taken from a to 0 instead of from 0 to a, and state u defined by (6)-(9) is the only one used.…”

Accuracy of electrocardiographic imaging using the method of fundamental solutions

“…The MFS approach has been applied to various inverse problems previously [26,27] and, as mentioned above, in particular to the inverse problem of electrocardiology [24]. Recent applications of the MFS to ECGI include studies of the locations of the heart and torso boundaries [28], application of the U-curve and the discrete Picard condition [29,30].…”