  The ability of convolutional neural networks (CNNs) to recognize objects
regardless of their position in the image is due to the
translation-equivariance of the convolutional operation. Group-equivariant CNNs
transfer this equivariance to other transformations of the input. Dealing
appropriately with objects and object parts of different scale is challenging,
and scale can vary for multiple reasons such as the underlying object size or
the resolution of the imaging modality. In this paper, we propose a
scale-equivariant convolutional network layer for three-dimensional data that
guarantees scale-equivariance in 3D CNNs. Scale-equivariance lifts the burden
of having to learn each possible scale separately, allowing the neural network
to focus on higher-level learning goals, which leads to better results and
better data-efficiency. We provide an overview of the theoretical foundations
and scientific work on scale-equivariant neural networks in the two-dimensional
domain. We then transfer the concepts from 2D to the three-dimensional space
and create a scale-equivariant convolutional layer for 3D data. Using the
proposed scale-equivariant layer, we create a scale-equivariant U-Net for
medical image segmentation and compare it with a non-scale-equivariant baseline
method. Our experiments demonstrate the effectiveness of the proposed method in
achieving scale-equivariance for 3D medical image analysis. We publish our code
at https://github.com/wimmerth/scale-equivariant-3d-convnet for further
research and application.
