Rotating coherent structures of water known as ocean eddies are the oceanic analog of storms in the atmosphere and a crucial component of ocean dynamics. In addition to dominating the ocean's kinetic energy, eddies play a significant role in the transport of water, salt, heat, and nutrients. Therefore, understanding current and future eddy activity is a central challenge to address future sustainability of marine ecosystems. The emergence of sea surface height observations from satellite radar altimeter has recently enabled researchers to track eddies at a global scale. The majority of studies that identify eddies from observational data employ highly parametrized connected component algorithms using expert filtered data, effectively making reproducibility and scalability challenging. In this paper, we improve upon the state-of-the-art connected component eddy monitoring algorithms to track eddies globally. This work makes three main contributions: first, we do not pre-process the data therefore minimizing the risk of wiping out important signals within the data. Second, we employ a physically-consistent convexity requirement on eddies based on theoretical and empirical studies to improve the accuracy and computational complexity of our method from quadratic to linear time in the size of each eddy. Finally, we accurately separate eddies that are in close spatial proximity, something existing methods cannot accomplish. We compare our results to those of the state of the art and discuss the impact of our improvements on the difference in results.