  This work introduces the Topological CNN (TCNN), which encompasses several
topologically defined convolutional methods. Manifolds with important
relationships to the natural image space are used to parameterize image filters
which are used as convolutional weights in a TCNN. These manifolds also
parameterize slices in layers of a TCNN across which the weights are localized.
We show evidence that TCNNs learn faster, on less data, with fewer learned
parameters, and with greater generalizability and interpretability than
conventional CNNs. We introduce and explore TCNN layers for both image and
video data. We propose extensions to 3D images and 3D video.
