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
https://github.com/xi-jia/Fourier-Net.

利用傅立叶网络替代资源密集型的 U-Net 风格扩张路径，我们提出了 Fourier-Net，通过学习位于有限带宽傅立叶域中的位移场的低维表示，然后通过模型驱动的解码器将其转换为空间域中的全分辨率位移场，达到减少计算成本、提高注册性能的效果。

Fourier-Net+: 基于有限频带表示的高效三维医学图像配准

Fourier-Net+: Leveraging Band-Limited Representation for Efficient 3D  Medical Image Registration

Unsupervised image registration commonly adopts U-Net style networks 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 tackle this problem, we propose the
Fourier-Net, replacing the expansive path in a U-Net style network with a
parameter-free model-driven decoder. Specifically, instead of our Fourier-Net
learning to output a full-resolution displacement field in the spatial domain,
we learn its low-dimensional representation in a band-limited Fourier domain.
This representation is then decoded by our devised model-driven decoder
(consisting of a zero padding layer and an inverse discrete Fourier transform
layer) to the dense, full-resolution displacement field in the spatial domain.
These changes allow our unsupervised Fourier-Net to contain fewer parameters
and computational operations, resulting in faster inference speeds. Fourier-Net
is then evaluated on two public 3D brain datasets against various
state-of-the-art approaches. For example, when compared to a recent
transformer-based method, i.e., TransMorph, our Fourier-Net, only using
0.22$\%$ of its parameters and 6.66$\%$ of the mult-adds, achieves a 0.6\%
higher Dice score and an 11.48$\times$ faster inference speed. Code is
available at https://github.com/xi-jia/Fourier-Net.

本文提出 Fourier-Net 网络，使用参数自由的基于模型的解码器代替 U-Net 风格网络的扩张路径。将低维度表示在带限傅里叶域中进行学习，最后由模型驱动的解码器解码为完整的空间域中的密集位移场。与现有方法相比，我们的 Fourier-Net 网络具有更少的参数和更快的推理速度，在公共 3D 脑数据集上实现了更高的 Dice 得分和更快的推理速度。