# CA-CGNet: Component-Aware Capsule Graph Neural Network for Non-Rigid Shape Correspondence
## 1 Introduction  
With the continuous development of computer 3D imaging technology, shape correspondence has become an important research direction in computer vision, which is closely related to object recognition, 3D reconstruction, image retrieval, image analysis, and other problems. The shape correspondence problem can be summarized as identifying homologous points of two or more shapes, where a large number of variables are needed to define a dense map for a non-rigid object due to its complex changes such as deformation, distortion, and extension. Although this problem has made some breakthroughs [1; 2; 3; 4], finding dense shape correspondence is still very challenging. Traditional non-rigid shape correspondence methods are generally established by measuring the similarity of point descriptors between two shapes, such as spectral geometry [5] or diffusion distance [6]. However, these methods rely heavily on initialization, which falls easily into local optimization with slow matching speed and low matching accuracy. Faced with large amounts of data and dense correspondence, scholars have further proposed to obtain the features of each point on the manifold surface through neural networks [7; 8; 9]. Unfortunately, these techniques directly regress the dense relationship of all input points, severely over-parameterizing the deformation and leading to poor generalization. Recent efforts mainly focus on functional map frameworks [10; 11; 12; 13] that converts point-to-point correspondence between models into a linear transformation in function space, mappinga point on the source model to multiple points on the target model to solve ambiguity problems. Despite significant progress in this area, these methods ignored the hierarchical geometric relations of the associated semantic parts. When the differences between the two input models are large, the extracted non-rigid shape descriptors lack sensitivity to specific information such as orientation and position, leading to unreliable correspondence results. Based on the above limitation, we propose a novel capsule graph neural network based on component constraint for the 3D non-rigid shape correspondence, called CA-CGNet, to further capture descriptors with powerful representation abilities. The entire framework of our method is given in Figure 1. The over-segmentation method is used to segment the mesh into several patches, which are clustered to form components using a dynamic clustering algorithm. In order to further improve the quality of feature acquisition, we design a novel graph routing algorithm to learn the relationship between the capsules in the same layer through a component-aware graph neural network (CA-GNN). Then, the knowledge distillation strategy is used to improve the convergence speed of the network by reducing the parameters while maintaining accuracy. Finally, a component pair constraint is added to the functional map with a component-based semantic loss function, which makes the shape correspondence more accurate and improves the robustness of the network. Our main contributions can be summarized as follows:  
* We design a novel network structure called CA-CGNet, which enhances the expressive ability of the network to represent object spatial pose and orientation changes by adding local mesh pose details to achieve high-precision shape correspondence. * To reduce the interference of noisy patch assignments, we propose a dynamic clustering algorithm to cluster the over-segmented meshes dynamically according to feature similarity to form components. By adding a component constraint to the functional maps and integrating component-based semantic constraint loss into the regularization term, the accuracy of shape correspondence is further improved. * The component-aware graph routing treats capsules as nodes in a graph neural network by adding component constraints to obtain more accurate relationships between capsules.