  U-Net style networks are commonly utilized in unsupervised image registration
to predict dense displacement fields, which for high-resolution volumetric
image data is a resource-intensive and time-consuming task. To tackle this
challenge, we first propose Fourier-Net, which replaces the costly U-Net style
expansive path with a parameter-free model-driven decoder. Instead of directly
predicting a full-resolution displacement field, our Fourier-Net learns a
low-dimensional representation of the displacement field in the band-limited
Fourier domain which our model-driven decoder converts to a full-resolution
displacement field in the spatial domain. Expanding upon Fourier-Net, we then
introduce Fourier-Net+, which additionally takes the band-limited spatial
representation of the images as input and further reduces the number of
convolutional layers in the U-Net style network's contracting path. Finally, to
enhance the registration performance, we propose a cascaded version of
Fourier-Net+. We evaluate our proposed methods on three datasets, on which our
proposed Fourier-Net and its variants achieve comparable results with current
state-of-the art methods, while exhibiting faster inference speeds, lower
memory footprint, and fewer multiply-add operations. With such small
computational cost, our Fourier-Net+ enables the efficient training of
large-scale 3D registration on low-VRAM GPUs. Our code is publicly available at
\url{https://github.com/xi-jia/Fourier-Net}.

Image and Video Processing;Computer Vision and Pattern Recognition