Deep learning methods have achieved great success in analyzing traditional data such as texts, sounds, images and videos. More and more research works are carrying out to extend standard deep learning technologies to geometric data such as point cloud or voxel grid of 3D objects, real life networks such as social and citation network. Many methods have been proposed in the research area. In this work, we aim to provide a comprehensive survey of geometric deep learning and related methods. First, we introduce the relevant knowledge and history of geometric deep learning field as well as the theoretical background. In the method part, we review different graph network models for graphs and manifold data. Besides, practical applications of these methods, datasets currently available in different research area and the problems and challenges are also summarized.

Objective. This study aims to enhance medical image registration by addressing the limitations of existing approaches that rely on spatial transformations through U-Net, ConvNets, or Transformers. The objective is to develop a novel architecture that combines ConvNets, graph neural networks (GNNs), and capsule networks to improve the accuracy and efficiency of medical image registration, which can also deal with the problem of rotating registration. Approach. We propose an deep learning-based approach which can be utilized in both unsupervised and semi-supervised manners, named as HGCMorph. It leverages a hybrid framework that integrates ConvNets and GNNs to capture lower-level features, specifically short-range attention, while also utilizing capsule networks (CapsNets) to model abstract higher-level features, including entity properties such as position, size, orientation, deformation, and texture. This hybrid framework aims to provide a comprehensive representation of anatomical structures and their spatial relationships in medical images. Main results. The results demonstrate the superiority of HGCMorph over existing state-of-the-art deep learning-based methods in both qualitative and quantitative evaluations. In unsupervised training process, our model outperforms the recent SOTA method TransMorph by achieving 7%/38% increase on Dice score coefficient (DSC), and 2%/7% improvement on negative jacobian determinant for OASIS and LPBA40 datasets, respectively. Furthermore, HGCMorph achieves improved registration accuracy in semi-supervised training process. In addition, when dealing with complex 3D rotations and secondary randomly deformations, our method still achieves the best performance. We also tested our methods on lung datasets, such as Japanese Society of Radiology, Montgoermy and Shenzhen. Significance. The significance lies in its innovative design to medical image registration. HGCMorph offers a novel framework that overcomes the limitations of existing methods by efficiently capturing both local and abstract features, leading to enhanced registration accuracy, discontinuity-preserving, and pose-learning abilities. The incorporation of capsule networks introduces valuable improvements, making the proposed method a valuable contribution to the field of medical image analysis. HGCMorph not only advances the SOTA methods but also has the potential to improve various medical applications that rely on accurate image registration.