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}.

翻译：U-Net架构网络在无监督图像配准中常被用于预测稠密位移场，但对于高分辨率体素图像数据而言，这是一项资源密集且耗时的任务。为解决这一挑战，我们首先提出Fourier-Net，该方法以无参数模型驱动解码器替代昂贵的U-Net扩展路径。我们的Fourier-Net不直接预测全分辨率位移场，而是在带限傅里叶域中学习位移场的低维表征，随后由模型驱动解码器将其转换为空间域中的全分辨率位移场。在Fourier-Net基础上，我们进一步提出Fourier-Net+，该方法额外将图像的带限空间表征作为输入，并进一步减少U-Net架构网络中收缩路径的卷积层数量。最后，为提升配准性能，我们提出级联版本的Fourier-Net+。我们在三个数据集上评估所提方法，实验表明Fourier-Net及其变体在取得与当前最先进方法相当结果的同时，展现出更快的推理速度、更低的内存占用和更少的乘法累加操作。凭借如此微小的计算成本，我们的Fourier-Net+能够在低显存GPU上高效训练大规模三维配准模型。我们的代码已开源：\url{https://github.com/xi-jia/Fourier-Net}。