  We introduce a method for training neural networks to perform image or volume
segmentation in which prior knowledge about the topology of the segmented
object can be explicitly provided and then incorporated into the training
process. By using the differentiable properties of persistent homology, a
concept used in topological data analysis, we can specify the desired topology
of segmented objects in terms of their Betti numbers and then drive the
proposed segmentations to contain the specified topological features.
Importantly this process does not require any ground-truth labels, just prior
knowledge of the topology of the structure being segmented. We demonstrate our
approach in three experiments. Firstly we create a synthetic task in which
handwritten MNIST digits are de-noised, and show that using this kind of
topological prior knowledge in the training of the network significantly
improves the quality of the de-noised digits. Secondly we perform an experiment
in which the task is segmenting the myocardium of the left ventricle from
cardiac magnetic resonance images. We show that the incorporation of the prior
knowledge of the topology of this anatomy improves the resulting segmentations
in terms of both the topological accuracy and the Dice coefficient. Thirdly, we
extend the method to 3D volumes and demonstrate its performance on the task of
segmenting the placenta from ultrasound data, again showing that incorporating
topological priors improves performance on this challenging task. We find that
embedding explicit prior knowledge in neural network segmentation tasks is most
beneficial when the segmentation task is especially challenging and that it can
be used in either a semi-supervised or post-processing context to extract a
useful training gradient from images without pixelwise labels.
