Synthesis and analysis models for sparse signal reconstruction in the inverse problem of ECG

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.