  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
