  Multi-class segmentation of cardiac magnetic resonance (CMR) images seeks a
separation of data into anatomical components with known structure and
configuration. The most popular CNN-based methods are optimised using pixel
wise loss functions, ignorant of the spatially extended features that
characterise anatomy. Therefore, whilst sharing a high spatial overlap with the
ground truth, inferred CNN-based segmentations can lack coherence, including
spurious connected components, holes and voids. Such results are implausible,
violating anticipated anatomical topology. In response, (single-class)
persistent homology-based loss functions have been proposed to capture global
anatomical features. Our work extends these approaches to the task of
multi-class segmentation. Building an enriched topological description of all
class labels and class label pairs, our loss functions make predictable and
statistically significant improvements in segmentation topology using a
CNN-based post-processing framework. We also present (and make available) a
highly efficient implementation based on cubical complexes and parallel
execution, enabling practical application within high resolution 3D data for
the first time. We demonstrate our approach on 2D short axis and 3D whole heart
CMR segmentation, advancing a detailed and faithful analysis of performance on
two publicly available datasets.
