  Segmentation networks are not explicitly imposed to learn global invariants
of an image, such as the shape of an object and the geometry between multiple
objects, when they are trained with a standard loss function. On the other
hand, incorporating such invariants into network training may help improve
performance for various segmentation tasks when they are the intrinsic
characteristics of the objects to be segmented. One example is segmentation of
aorta and great vessels in computed tomography (CT) images where vessels are
found in a particular geometry in the body due to the human anatomy and they
mostly seem as round objects on a 2D CT image. This paper addresses this issue
by introducing a new topology-aware loss function that penalizes topology
dissimilarities between the ground truth and prediction through persistent
homology. Different from the previously suggested segmentation network designs,
which apply the threshold filtration on a likelihood function of the prediction
map and the Betti numbers of the ground truth, this paper proposes to apply the
Vietoris-Rips filtration to obtain persistence diagrams of both ground truth
and prediction maps and calculate the dissimilarity with the Wasserstein
distance between the corresponding persistence diagrams. The use of this
filtration has advantage of modeling shape and geometry at the same time, which
may not happen when the threshold filtration is applied. Our experiments on
4327 CT images of 24 subjects reveal that the proposed topology-aware loss
function leads to better results than its counterparts, indicating the
effectiveness of this use.
