Medical image registration plays an important role in clinical applications such as surgical navigation，disease progression monitoring，and multimodal diagnosis. The accuracy of image registration is critical to ensuring the reliability of subsequent image analysis. Traditional methods formulate the registration process as an optimization problem by maximizing the similarity between the fixed and warped moving images and employ iterative optimization strategies to update the deformation field gradually. While such approaches provide clear theoretical interpretability，their high computational costs fail to meet real-time clinical demands. By contrast，deep learning-based methods leverage large-scale datasets to learn the deformation field directly. Although these data-driven approaches have demonstrated remarkable improvements in registration speed and accuracy，they operate as black-box models with limited interpretability. Moreover，most current approaches continue to depend on manually designed regularization terms，which may lack adaptability to diverse anatomical structures. To address these challenges，we propose an unfolding-based registration network that combines the theoretical guarantees of traditional optimization with the high efficiency and adaptive learning capabilities of deep neural networks.

Specifically，by introducing variable splitting，we first decouple the registration problem into two subproblems： a similarity-constraint subproblem and a deformation-regularization subproblem. This decomposition reduces optimization complexity and allows each subproblem to be addressed separately. Inspired by deep unfolding networks，the method maps the alternating update process between the two subproblems to a multistage cascaded architecture. Each stage corresponds to one complete iteration of the optimizer，providing clear physical and mathematical foundations and enhancing interpretability compared with traditional end-to-end models. Within this framework，two specialized modules are designed to address each subproblem. The fidelity module （FM） is employed to solve the similarity-constraint subproblem. Convolutional neural networks （CNN） are widely used in deep learning-based registration. However，the limited receptive field of the convolution kernel leads the network to focus on local feature information，while the Transformer uses the self-attention mechanism to capture global dependencies effectively. By integrating the strengths of these two architectures，the FM is designed on the basis of the TransUnet framework. This hybrid design allows the FM to combine local features with global context. The denoising module （DM） is designed to solve the deformation-regularization subproblem. Instead of using handcrafted regularization terms，we approximate the regularization term using a trainable neural network. Concretely，the DM adopts the denoising convolutional neural network architecture and incorporates residual connections，allowing the model to learn priors adaptively in a data-driven manner and thus improving generalization across different anatomical structures. These two modules jointly complete a single alternating iteration and are cascaded to form the deep unfolding network. The overall network is trained with a loss function that balances image similarity，deformation consistency between the outputs of the FM and DM，and smoothness of the deformation field. The proposed network is implemented using the PyTorch framework and trained on an NVIDIA Tesla V100 GPU with 32 GB of memory. The model is optimized using the Adam optimizer with a learning rate of 1E-4，a batch size of 1，and is trained for 500 epochs. The Transformer component in the FM was configured with four attention heads，eight encoder layers，and an MLP dimension of 96. The unfolding network was constructed with three cascaded stages.

Experiments were conducted on two publicly available brain magnetic resonance imaging （MRI） datasets： LPBA40 and OASIS. We compared our method with a range of state-of-the-art models，including traditional registration methods （i.e.，large deformation diffeomorphic metric mapping and symmetric normalization） and learning-based methods （e.g.，VoxelMorph，TransMorph，UTSRMorph and H-SGANet）. The quantitative evaluation metrics contained a dice similarity coefficient （Dice），the percentage of nonpositive Jacobian determinants，and inference time. The proposed method achieved an average Dice of 70.1% on LPBA40 and 79.9% on OASIS，while the percentage of nonpositive Jacobian determinants remained at a low level of 0.01% on both datasets，and the inference time was 0.55 s on LPBA40 and 0.37 s on OASIS. Although slightly higher than some deep learning-based models due to the cascaded unfolding architecture and Transformer modules，the time remained clinically feasible. Visualized results also indicated that the deformation fields produced by the proposed method was smoother and more consistent with anatomical priors. Additionally，we conducted a set of ablation studies and reported corresponding quantitative results. These ablation studies explicitly demonstrated the effectiveness of individual modules in our proposed model，including the variable splitting strategy，the TransUnet-based fidelity module，and the residual connection-based denoising module and assessed the influence of key hyperparameters.

In this study，we proposed a deep unfolding network for medical image registration that effectively combines the interpretability of traditional iterative methods with the efficiency and adaptability of deep learning. By unrolling a decoupled optimization model into a trainable network with fidelity and denoising modules，the proposed approach strictly corresponds to a specific step in iterative algorithms. Experiments demonstrate that the proposed method remarkably enhances interpretability while maintaining high efficiency and accuracy.