# Bridging Classical and Learning-based Iterative Registration through Deep Equilibrium Models

arXiv: https://arxiv.org/abs/2507.00582 · HTML: https://mlsanity.com/paper/2507.00582/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 27 references that are in the map.
- First posted on arXiv 1 Jul 2025.

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

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network](https://mlsanity.com/paper/1808.03314/) (2018)
- [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)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Learn2Reg: comprehensive multi-task medical image registration challenge, dataset and evaluation in the era of deep learning](https://mlsanity.com/paper/2112.04489/) (2021)
- [Deep Learning in Medical Image Registration: A Survey](https://mlsanity.com/paper/1903.02026/) (2019)
- [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)
- [On Training Implicit Models](https://mlsanity.com/paper/2111.05177/) (2021)
- [Deep Equilibrium Optical Flow Estimation](https://mlsanity.com/paper/2204.08442/) (2022)
- [mlVIRNET: Multilevel Variational Image Registration Network](https://mlsanity.com/paper/1909.10084/) (2019)
- [TorchDEQ: A Library for Deep Equilibrium Models](https://mlsanity.com/paper/2310.18605/) (2023)
- [Embedding Gradient-based Optimization in Image Registration Networks](https://mlsanity.com/paper/2112.03915/) (2021)
- [A Plug-and-Play Image Registration Network](https://mlsanity.com/paper/2310.04297/) (2023)
- [Recurrent Inference Machine for Medical Image Registration](https://mlsanity.com/paper/2406.13413/) (2024)

## Built on by (most cited first, 2 of 2)

- [Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach](https://mlsanity.com/paper/2602.03297/) (2026)
- [Adapting Frozen Mono-modal Backbones for Multi-modal Registration via Contrast-Agnostic Instance Optimization](https://mlsanity.com/paper/2603.26393/) (2026)

## Abstract (arXiv)

Deformable medical image registration is traditionally formulated as an optimization problem. While classical methods solve this problem iteratively, recent learning-based approaches use recurrent neural networks (RNNs) to mimic this process by unrolling the prediction of deformation fields in a fixed number of steps. However, classical methods typically converge after sufficient iterations, but learning-based unrolling methods lack a theoretical convergence guarantee and show instability empirically. In addition, unrolling methods have a practical bottleneck at training time: GPU memory usage grows linearly with the unrolling steps due to backpropagation through time (BPTT). To address both theoretical and practical challenges, we propose DEQReg, a novel registration framework based on Deep Equilibrium Models (DEQ), which formulates registration as an equilibrium-seeking problem, establishing a natural connection between classical optimization and learning-based unrolling methods. DEQReg maintains constant memory usage, enabling theoretically unlimited iteration steps. Through extensive evaluation on the public brain MRI and lung CT datasets, we show that DEQReg can achieve competitive registration performance, while substantially reducing memory consumption compared to state-of-the-art unrolling methods. We also reveal an intriguing phenomenon: the performance of existing unrolling methods first increases slightly then degrades irreversibly when the inference steps go beyond the training configuration. In contrast, DEQReg achieves stable convergence with its inbuilt equilibrium-seeking mechanism, bridging the gap between classical optimization-based and modern learning-based registration methods.