# Fourier-Net: Fast Image Registration with Band-limited Deformation

arXiv: https://arxiv.org/abs/2211.16342 · HTML: https://mlsanity.com/paper/2211.16342/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 35 references that are in the map.
- First posted on arXiv 29 Nov 2022.

## Builds on (most cited first, 18 of 18)

- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [A Deep Learning Framework for Unsupervised Affine and Deformable Image Registration](https://mlsanity.com/paper/1809.06130/) (2018)
- [Large Deformation Diffeomorphic Image Registration with Laplacian Pyramid Networks](https://mlsanity.com/paper/2006.16148/) (2020)
- [Learn2Reg: comprehensive multi-task medical image registration challenge, dataset and evaluation in the era of deep learning](https://mlsanity.com/paper/2112.04489/) (2021)
- [Unsupervised Learning for Fast Probabilistic Diffeomorphic Registration](https://mlsanity.com/paper/1805.04605/) (2018)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [Fast Symmetric Diffeomorphic Image Registration with Convolutional Neural Networks](https://mlsanity.com/paper/2003.09514/) (2020)
- [Recursive Cascaded Networks for Unsupervised Medical Image Registration](https://mlsanity.com/paper/1907.12353/) (2019)
- [HyperMorph: Amortized Hyperparameter Learning for Image Registration](https://mlsanity.com/paper/2101.01035/) (2021)
- [CycleMorph: Cycle Consistent Unsupervised Deformable Image Registration](https://mlsanity.com/paper/2008.05772/) (2020)
- [U-Net vs Transformer: Is U-Net Outdated in Medical Image Registration?](https://mlsanity.com/paper/2208.04939/) (2022)
- [DeepFLASH: An Efficient Network for Learning-based Medical Image Registration](https://mlsanity.com/paper/2004.02097/) (2020)
- [Inverse-Consistent Deep Networks for Unsupervised Deformable Image Registration](https://mlsanity.com/paper/1809.03443/) (2018)
- [Learning a Model-Driven Variational Network for Deformable Image Registration](https://mlsanity.com/paper/2105.12227/) (2021)
- [Nesterov Accelerated ADMM for Fast Diffeomorphic Image Registration](https://mlsanity.com/paper/2109.12688/) (2021)
- [An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration](https://mlsanity.com/paper/1408.6299/) (2014)

## Built on by (most cited first, 22 of 22)

- [A survey on deep learning in medical image registration: new technologies, uncertainty, evaluation metrics, and beyond](https://mlsanity.com/paper/2307.15615/) (2023)
- [Spatially Covariant Image Registration with Text Prompts](https://mlsanity.com/paper/2311.15607/) (2023)
- [3D-GSW: 3D Gaussian Splatting for Robust Watermarking](https://mlsanity.com/paper/2409.13222/) (2024)
- [Fourier Prompt Tuning for Modality-Incomplete Scene Segmentation](https://mlsanity.com/paper/2401.16923/) (2024)
- [Slicer Networks](https://mlsanity.com/paper/2401.09833/) (2024)
- [Decoder-Only Image Registration](https://mlsanity.com/paper/2402.03585/) (2024)
- [Fourier-Net+: Leveraging Band-Limited Representation for Efficient 3D Medical Image Registration](https://mlsanity.com/paper/2307.02997/) (2023)
- [WiNet: Wavelet-based Incremental Learning for Efficient Medical Image Registration](https://mlsanity.com/paper/2407.13426/) (2024)
- [EfficientMorph: Parameter-Efficient Transformer-Based Architecture for 3D Image Registration](https://mlsanity.com/paper/2403.11026/) (2024)
- [Encoder-Only Image Registration](https://mlsanity.com/paper/2509.00451/) (2025)
- [SACB-Net: Spatial-awareness Convolutions for Medical Image Registration](https://mlsanity.com/paper/2503.19592/) (2025)
- [VoxelOpt: Voxel-Adaptive Message Passing for Discrete Optimization in Deformable Abdominal CT Registration](https://mlsanity.com/paper/2506.19975/) (2025)
- [SAMIR, an efficient registration framework via robust feature learning from SAM](https://mlsanity.com/paper/2509.13629/) (2025)
- [4DLoG: Generative Modeling of Neurodegenerative Brain Anatomy with 4D Longitudinal Diffusion Model](https://mlsanity.com/paper/2604.22700/) (2026)
- [OncoReg: Medical Image Registration for Oncological Challenges](https://mlsanity.com/paper/2503.23179/) (2025)
- [An Efficient Model-Driven Groupwise Approach for Atlas Construction](https://mlsanity.com/paper/2508.10743/) (2025)
- [Ideal Registration? Segmentation is All You Need](https://mlsanity.com/paper/2509.15784/) (2025)
- [Unsupervised Deformable Image Registration with Local-Global Attention and Image Decomposition](https://mlsanity.com/paper/2601.14337/) (2026)
- [Reg-TTR, Test-Time Refinement for Fast, Robust and Accurate Image Registration](https://mlsanity.com/paper/2601.19114/) (2026)
- [Flow Matching-enabled Test-Time Refinement for Unsupervised Cardiac MR Registration](https://mlsanity.com/paper/2603.01073/) (2026)
- [Low-Rank Velocity Fields as a Structural Prior for Unsupervised 4D Medical Image Interpolation](https://mlsanity.com/paper/2608.24025/) (2026)
- [Fast and Accurate Monomodal 3D High Resolution Deep Registration of Drosophila Larval Brain Volumes](https://mlsanity.com/paper/2609.11240/) (2026)

## Abstract (arXiv)

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, named TransMorph, our Fourier-Net, which only uses 2.2% of its parameters and 6.66% of the multiply-add operations, achieves a 0.5% higher Dice score and an 11.48 times faster inference speed. Code is available at urlhttps://github.com/xi-jia/Fourier-Net.