RotaTouille: Rotation Equivariant Deep Learning for Contours
Authors: name: Odin Hoff Gardaa, name: Nello Blaser

Contours or closed planar curves are common in many domains. For example,
they appear as object boundaries in computer vision, isolines in meteorology,
and the orbits of rotating machinery. In many cases when learning from contour
data, planar rotations of the input will result in correspondingly rotated
outputs. It is therefore desirable that deep learning models be rotationally
equivariant. In addition, contours are typically represented as an ordered
sequence of edge points, where the choice of starting point is arbitrary. It is
therefore also desirable for deep learning methods to be equivariant under
cyclic shifts. We present RotaTouille, a deep learning framework for learning
from contour data that achieves both rotation and cyclic shift equivariance
through complex-valued circular convolution. We further introduce and
characterize equivariant non-linearities, coarsening layers, and global pooling
layers to obtain invariant representations for downstream tasks. Finally, we
demonstrate the effectiveness of RotaTouille through experiments in shape
classification, reconstruction, and contour regression.