  Group convolutional neural networks (G-CNNs) can be used to improve classical
CNNs by equipping them with the geometric structure of groups. Central in the
success of G-CNNs is the lifting of feature maps to higher dimensional
disentangled representations, in which data characteristics are effectively
learned, geometric data-augmentations are made obsolete, and predictable
behavior under geometric transformations (equivariance) is guaranteed via group
theory. Currently, however, the practical implementations of G-CNNs are limited
to either discrete groups (that leave the grid intact) or continuous compact
groups such as rotations (that enable the use of Fourier theory). In this paper
we lift these limitations and propose a modular framework for the design and
implementation of G-CNNs for arbitrary Lie groups. In our approach the
differential structure of Lie groups is used to expand convolution kernels in a
generic basis of B-splines that is defined on the Lie algebra. This leads to a
flexible framework that enables localized, atrous, and deformable convolutions
in G-CNNs by means of respectively localized, sparse and non-uniform B-spline
expansions. The impact and potential of our approach is studied on two
benchmark datasets: cancer detection in histopathology slides in which rotation
equivariance plays a key role and facial landmark localization in which scale
equivariance is important. In both cases, G-CNN architectures outperform their
classical 2D counterparts and the added value of atrous and localized group
convolutions is studied in detail.
