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Spatio-Temporal Solutions in Inverse Electrocardiography

Conventional electrocardiography (ECG) is an essential tool for investigating cardiac disorders such as arrhythmias or myocardial infarction. It involves interpretation of potentials recorded at the body surface that occur due to the electrical activity of the heart. Estimation of epicardial potentials from these recorded signals is known as inverse problem of ECG. It is difficult to solve this problem for effective cardiac imaging due to the ill-posed nature and high dimensionality of the problem. There are many solution approaches in order to cope with these difficulties. We used Tikhonov regularization and Bayesian Maximum A Posteriori (MAP) estimation methods in this study. The traditional approach is to solve the problem at each time instant separately (column sequential approach). This is the fastest and easiest approach; however it does not include temporal correlations of the epicardial potentials. Greensite (2002) proposed certain specific assumptions about structures inherent in the problem formulation that allow the use of spatial and temporal constraints simultaneously. In this study, we applied this framework to solve the spatiotemporal inverse ECG problem. We first applied a temporal whitening filter to the original problem. The new set of equations, which became temporally decorrelated, was solved using the column sequential approach. The desired solution was obtained by transforming the result back into the original domain. In the spatiotemporal Bayesian MAP approach, the covariance matrix also changes after the application of the whitening transformation. We also derived an expression for the new covariance matrix in terms of the transformation matrix and the original covariance matrix of the epicardial potentials.