Improving Tikhonov regularization with linearly constrained optimization: application to the inverse epicardial potential solution

Two methods to improve on the accuracy of the Tikhonov regularization technique commonly used for the stable recovery of solutions to ill-posed problems are presented. These methods do not require a priori knowledge of the properties of the solution or of the error. Rather they exploit the observed properties of overregularized and underregularized Tikhonov solutions so as to impose linear constraints on the sought-after solution. The two methods were applied to the inverse problem of electrocardiography using a spherical heart-torso model and simulated inner-sphere (epicardial) and outer-sphere (body) potential distributions. It is shown that if the overregularized and underregularized Tikhonov solutions are chosen properly, the two methods yield epicardial solutions that are not only more accurate than the optimal Tikhonov solution but also provide other qualitative information, such as correct position of the extrema, not obtainable using ordinary Tikhonov regularization. A heuristic method to select the overregularized and underregularized solutions is discussed.