The goal of ECG-imaging (ECGI) is to reconstruct heart electrical activity from body surface potential maps. The problem is ill-posed, which means that it is extremely sensitive to measurement and modeling errors. The most commonly used method to tackle this obstacle is Tikhonov regularization, which consists in converting the original problem into a well-posed one by adding a penalty term. The method, despite all its practical advantages, has however a serious drawback: The obtained solution is often over-smoothed, which can hinder precise clinical diagnosis and treatment planning. In this paper, we apply a binary optimization approach to the transmembrane voltage (TMV)-based problem. For this, we assume the TMV to take two possible values according to a heart abnormality under consideration. In this work, we investigate the localization of simulated ischemic areas and ectopic foci and one clinical infarction case. This affects only the choice of the binary values, while the core of the algorithms remains the same, making the approximation easily adjustable to the application needs. Two methods, a hybrid metaheuristic approach and the difference of convex functions (DC), algorithm were tested. For this purpose, we performed realistic heart simulations for a complex thorax model and applied the proposed techniques to the obtained ECG signals. Both methods enabled localization of the areas of interest, hence showing their potential for application in ECGI. For the metaheuristic algorithm, it was necessary to subdivide the heart into regions in order to obtain a stable solution unsusceptible to the errors, while the analytical DC scheme can be efficiently applied for higher dimensional problems. With the DC method, we also successfully reconstructed the activation pattern and origin of a simulated extrasystole. In addition, the DC algorithm enables iterative adjustment of binary values ensuring robust performance.
Electrocardiographic imaging (ECGI) is a non-invasive diagnostical tool solving the inverse problem of ECG, which means the reconstruction of electrical potentials in the heart from the ECG data. The ill-posednees of this problem makes necessary addition of a-priori information. A typical approach is the Tikhonov regularization looking for the best balance between minimizing the data misfit and the regularization term which characterizes desired properties of the solution. However, the quality of an obtained solution, and as a result its clinical relevance, could be significantly improved by application of methods for non-smooth regularization. In this work we introduced a possible dictionary definition for the electrical sources in the heart: we subdivided the heart into 100 pieces and considered them to constitute the columns of our dictionary. We also provided a short discussion on differences between synthesis and analysis models, tested the analysis algorithm with a penalty matrix which is not related to the defined dictionary (discrete gradient operator for all heart points) and compared the performance of these three algorithms for two simulated ventricular ectopic foci. The analysis method with the gradient operator showed a slightly superior performance although all methods correctly identified the regions of interest.