  Reconstructing 3D objects from 2D images is both challenging for our brains
and machine learning algorithms. To support this spatial reasoning task,
contextual information about the overall shape of an object is critical.
However, such information is not captured by established loss terms (e.g. Dice
loss). We propose to complement geometrical shape information by including
multi-scale topological features, such as connected components, cycles, and
voids, in the reconstruction loss. Our method uses cubical complexes to
calculate topological features of 3D volume data and employs an optimal
transport distance to guide the reconstruction process. This topology-aware
loss is fully differentiable, computationally efficient, and can be added to
any neural network. We demonstrate the utility of our loss by incorporating it
into SHAPR, a model for predicting the 3D cell shape of individual cells based
on 2D microscopy images. Using a hybrid loss that leverages both geometrical
and topological information of single objects to assess their shape, we find
that topological information substantially improves the quality of
reconstructions, thus highlighting its ability to extract more relevant
features from image datasets.
