  Analyzing volumetric data with rotational invariance or equivariance is an
active topic in current research. Existing deep-learning approaches utilize
either group convolutional networks limited to discrete rotations or steerable
convolutional networks with constrained filter structures. This work proposes a
novel equivariant neural network architecture that achieves analytical
Equivariance to Local Pattern Orientation on the continuous SO(3) group while
allowing unconstrained trainable filters - EquiLoPO Network. Our key
innovations are a group convolutional operation leveraging irreducible
representations as the Fourier basis and a local activation function in the
SO(3) space that provides a well-defined mapping from input to output
functions, preserving equivariance. By integrating these operations into a
ResNet-style architecture, we propose a model that overcomes the limitations of
prior methods. A comprehensive evaluation on diverse 3D medical imaging
datasets from MedMNIST3D demonstrates the effectiveness of our approach, which
consistently outperforms state of the art. This work suggests the benefits of
true rotational equivariance on SO(3) and flexible unconstrained filters
enabled by the local activation function, providing a flexible framework for
equivariant deep learning on volumetric data with potential applications across
domains. Our code is publicly available at
\url{https://gricad-gitlab.univ-grenoble-alpes.fr/GruLab/ILPO/-/tree/main/EquiLoPO}.
