Book excerpt: Solutions to the inverse problem of electrocardiography, also known as Electrocardiographic Imaging (ECGI), non-invasively image the electrical activity of the heart to localize and quantify normal and abnormal cardiac electrophysiology, and have recently attracted considerable attention in the research community. These methods have great promise to aid planning of catheter ablation procedures as well as for screening and diagnosis. They solve a problem that uses electrocardiographic measurements on the body surface (ECGs) to characterize the unknown electrical activity of the heart, here described by the time series of distribution of electrical potential on the heart surface. To do so, they rely on the calculation of a "forward" model relating the potentials on the heart with the ECGs, typically known as "forward matrix". However, ECGI has some limitations that impede its application to routine clinical practice. One of them is the ill-posedness of the forward model, which causes inverse solutions to be unreliable even with small levels of noise in the inputs and the model. One way of improving the resilience to this problem is to increase the Signal to Noise Ratio (SNR) of the signal by the incorporation of recordings from multiple heartbeats. During the course of this thesis we have addressed this approach in several applications: imaging the presence of T-wave alternans on the heart, localizing the site of earliest activation of pre-ventricular contractions (PVC) and the characterization of the errors introduced in the forward matrix. T-wave alternans (TWA), defined as a beat-to-beat alternation of the T-wave in the ECG, have been advocated for the past two decades as a promising marker of susceptibility to sudden cardiac death (SCD). However current ECG tests for TWA have been reported to have high sensitivity but low specificity for SCD and thus there is still no consensus on their predictive value. The distinction between concordant alternans---for which the whole heart alternates at the same phase---and discordant alternans---for which different regions of the heart alternate in different magnitude and phase---is of special interest as the latter have been observed preceding ventricular fibrillation in animal experiments and simulations. Thus, better characterization of the spatial behavior of discordant alternans could lead to improvements in the predictive value of TWA tests. Motivated by the desire to identify and localize discordant regions from body surface measurements, we present two studies related to TWA. In the first we introduce a theoretical characterization of how concordant and discordant alternans appear on the body surface. In the second study we present a method that directly estimates the location and phase of discordant TWA on the heart surface from body surface measurements by combining techniques from TWA detection with the methodology of ECGI. The results obtained with both studies on TWA suggest considerable promise that these methods may help increase our ability to understanding TWA based on body surface recordings, and thereby more clearly study its link to SCD, by providing previously unavailable spatial information to researchers and clinicians. Accurate detection of the site of first activation in a pre-ventricular contraction (PVC) can lead to substantial improvements in pre-procedure planning for ablation interventions and the subsequent reduction length and costs of these interventions. However, to be able to accurately localize these sites, it is necessary to obtain reliable inverse reconstructions of the activation wavefront that propagates throughout the heart during QRS. In this thesis, we present two approaches to increase the SNR through the incorporation of multiple recordings from the same underlying PVC activity. In the first one, we study deterministic averaging approaches to incorporate multiple heartbeats in an inverse solver. In the second one, we extend the previous averaging approaches into a statistically inspired model to characterize the beat-to-beat fluctuations observed on the ECG. These studies showed some improvement over regular averaging techniques, but the beat-to-beat variability still present in the solutions suggested that there is need to better characterize the sources of noise, which led to the third contribution of this thesis. Model mismatch in the forward matrix is a significant factor in inverse solution error. This is particularly problematic when incorporating multiple heartbeat recordings, as the heart moves within the thorax from beat to beat due to respiration. Therefore, the use of a single forward matrix necessarily introduces model errors in this setting. To overcome this limitation, we propose in this work to efficiently characterize the changes in the forward matrix due to movements of the heart and then adapt the forward matrix to each individual heartbeat. With that objective we developed a new method to characterize errors in the forward matrix that are produced by movement of the heart. This method approximates the changes in volume conductor geometry with a parameterized transform and then encapsulates the forward matrices that these changes would generate into a single mapping function that describes the whole sequence of computations. With this mapping, it is then possible to apply optimization methods that correct for the errors in the forward matrix, generating an approximate forward matrix corresponding to an estimated position and orientation of the heart and, if desired, also solve for the unknown cardiac electrical potentials in an iterative fashion. We expect this work to impact ECGI research and beyond. In ECGI, we expect to improve results in animal models used for understanding mechanisms and validating inverse solutions and to reduce uncertainty of inverse solutions obtained in clinical applications, and to expand the range of potential clinical problems for which ECGI can be reasonably applied by reducing the need for extensive high resolution anatomical imaging. Beyond ECGI, the capacity to non-invasively track the position of the heart has the potential to impact a number of other clinical problems, for example, improving catheter registration in ablation procedures.