  With respect to spatial overlap, CNN-based segmentation of short axis
cardiovascular magnetic resonance (CMR) images has achieved a level of
performance consistent with inter observer variation. However, conventional
training procedures frequently depend on pixel-wise loss functions, limiting
optimisation with respect to extended or global features. As a result, inferred
segmentations can lack spatial coherence, including spurious connected
components or holes. Such results are implausible, violating the anticipated
topology of image segments, which is frequently known a priori. Addressing this
challenge, published work has employed persistent homology, constructing
topological loss functions for the evaluation of image segments against an
explicit prior. Building a richer description of segmentation topology by
considering all possible labels and label pairs, we extend these losses to the
task of multi-class segmentation. These topological priors allow us to resolve
all topological errors in a subset of 150 examples from the ACDC short axis CMR
training data set, without sacrificing overlap performance.
