We present a novel methodology that combines graph and dense segmentation techniques by jointly learning both point and pixel contour representations, thereby leveraging the benefits of each approach. This addresses deficiencies in typical graph segmentation methods where misaligned objectives restrict the network from learning discriminative vertex and contour features. Our joint learning strategy allows for rich and diverse semantic features to be encoded, while alleviating common contour stability issues in dense-based approaches, where pixel-level objectives can lead to anatomically implausible topologies. In addition, we identify scenarios where correct predictions that fall on the contour boundary are penalised and address this with a novel hybrid contour distance loss. Our approach is validated on several Chest X-ray datasets, demonstrating clear improvements in segmentation stability and accuracy against a variety of dense- and point-based methods. Our source code is freely available at: www.github.com/kitbransby/Joint_Graph_Segmentation
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