## 1 Introduction  
Using medical images to diagnose and treat diseases has always been the main development direction in the field of medical image processing. Deformable registration techniques for brain images have found many medical applications, such as brain tumor localization, neurosurgery assistance, and treatment response assessment. The aim of deformable registration is to establish a set of optimal spatial correspondence between a fixed image and a moving image in the same space, so that the difference regions between the two images tend to be aligned. According to different medical scenarios, it is possible to perform atlas-to-patient registration, patient-to-patient registration, multimodal image registration, multi-view image registration, etc. At the same time, the results of image registration can also be applied to many other medical image processing techniques, such as image segmentation, image fusion, etc. Recently, methods based on deep learning have been widely used in the field of image analysis and processing, and they have demonstrated advantages such as improved accuracy, faster processing speed, and the ability to handle complex deformations, surpassing traditional algorithms. Since AlexNet [1] achieved outstanding performance, the convolutional neural network (CNN) has become one of the most successful models in deep learning. CNN has also made a major breakthrough in the field of medical image registration by virtue of its ability to efficiently process highly structured data. The CNN-based medical image registration methods can be divided into two categories: one is the iterative registration method using CNN for similarity prediction, and the other is the method of using CNN for transformation parameter prediction. The former can be called the method based on deep similarity, which mainly uses CNN to measure the similarity between multimodal images. Traditional intensity-based similarity measures such as cross-correlation (CC), square sum distance (SSD), and mean square distance (MSD) are very effective for the registration of single-modality images with the same intensity distribution. CC measures the linear correlation between two images, SSD calculates the sum of squared differences between corresponding pixels, and MSD is the average of squared differences, providing a robust measure of similarity. But for multimodal image registration, the performance of traditional similarity measures is not satisfactory. Consequently, some researchers have proposed using CNNs to measure the similarity between images, which has yielded promising results. However, these methods [2; 3; 4] still rely on traditional image registration algorithms for iterative optimization, which results in lower registration efficiency. The second category includes both supervised [5; 6; 7] and unsupervised [8; 9] learning-based registration approaches. Unlike the methods based on deep similarity, the above two methods infer the transformation parameters at one time in the forward prediction (a direct inference process that predicts transformation parameters without iterative optimization), which greatly improves the registration efficiency. The difference between these two methods is that the supervised models need some label information during the training to help the model to learn. However, it is usually difficult to obtain these required real information or ground truth. These data often need to be manually marked by experts [10] or generated by traditional registration algorithms [11; 12] or by random transformation [13]. The ground truth obtained by these methods may also be inconsistent with actual physiological changes, as it cannot always reflect real-world biological variability. In addition, these data will also limit the upper limit of registration. In contrast, unsupervised learning retains the benefits of supervised learning while not being constrained by the need for ground truth, making it increasingly favored by researchers.