Fourier-Net+: Band-Limited Spatial Representation for Efficient Medical Image Registration

U-Net style networks are commonly utilized in unsupervised image registration to predict dense displacement fields in the full-resolution spatial domain. For high-resolution volumetric image data, this process is, however, resource-intensive and time-consuming. To address this challenge, we propose Fourier-Net+, an image-domain deformable registration framework that operates on real-valued images for broad modality compatibility. Fourier-Net+ uses deterministic Fourier-domain band-limiting for efficient down- and up-sampling and employs a parameter-free, model-driven decoder to learn a band-limited, low-dimensional representation of the displacement field; all learnable network layers are real-valued. In addition, to enhance the registration performance and encourage diffeomorphism, we propose the cascaded and diffeomorphic versions of Fourier-Net+. We evaluate the proposed methods on five datasets, including two brain MRI, one 3-D cardiac MRI (3-D-CMR), one abdominal CT-MR dataset, and one noisy ultrasound cardiac dataset, comparing them against various state-of-the-art approaches. Our Fourier-Net+ and its variants achieve comparable results with these approaches while exhibiting faster training and inference speeds with a lower memory footprint and fewer multiply add operations (mult-adds). For example, on the 3-D-CMR dataset, our Diff-Fourier-Net+ significantly outperforms strong baselines such as TransMorph, TransMatch, and SACB-Net in Dice and HD, as well as in clinical metrics including end-diastolic (ED) volume and ejection fraction estimation, while using substantially less memory and computational cost. This efficiency enables large-scale 3-D registration training on low-VRAM GPUs.