  For many years, it has been shown how much exploiting equivariances can be
beneficial when solving image analysis tasks. For example, the superiority of
convolutional neural networks (CNNs) compared to dense networks mainly comes
from an elegant exploitation of the translation equivariance. Patterns can
appear at arbitrary positions and convolutions take this into account to
achieve translation invariant operations through weight sharing. Nevertheless,
images often involve other symmetries that can also be exploited. It is the
case of rotations and reflections that have drawn particular attention and led
to the development of multiple equivariant CNN architectures. Among all these
methods, Bessel-convolutional neural networks (B-CNNs) exploit a particular
decomposition based on Bessel functions to modify the key operation between
images and filters and make it by design equivariant to all the continuous set
of planar rotations. In this work, the mathematical developments of B-CNNs are
presented along with several improvements, including the incorporation of
reflection and multi-scale equivariances. Extensive study is carried out to
assess the performances of B-CNNs compared to other methods. Finally, we
emphasize the theoretical advantages of B-CNNs by giving more insights and
in-depth mathematical details.
