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Three-dimensional (3-D) data have many applications in the field of computer vision and a point cloud is one of the most popular modalities. Therefore, how to establish a good representation for a point cloud is a core issue in computer vision, especially for 3-D object recognition tasks. Existing approaches mainly focus on the invariance of representation under the group of permutations. However, for point cloud data, it should also be rotation invariant. To address such invariance, in this article, we introduce a relation of equivalence under the action of rotation group, through which the representation of point cloud is located in a homogeneous space. That is, two point clouds are regarded as equivalent when they are only different from a rotation. Our network is flexibly incorporated into existing frameworks for point clouds, which guarantees the proposed approach to be rotation invariant. Besides, a sufficient analysis on how to parameterize the group SO(3) into a convolutional network, which captures a relation with all rotations in 3-D Euclidean space $\mathbb {R}^{3}$. We select the optimal rotation as the best representation of point cloud and propose a solution for minimizing the problem on the rotation group SO(3) by using its geometric structure. To validate the rotation invariance, we combine it with two existing deep models and evaluate them on ModelNet40 dataset and its subset ModelNet10. Experimental results indicate that the proposed strategy improves the performance of those existing deep models when the data involve arbitrary rotations. [ABSTRACT FROM AUTHOR]

Effective 3D shape retrieval and recognition are challenging but important tasks in computer vision research field, which have attracted much attention in recent decades. Although recent progress has shown significant improvement of deep learning methods on 3D shape retrieval and recognition performance, it is still under investigated of how to jointly learn an optimal representation of 3D shapes considering their relationships. To tackle this issue, we propose a multi-scale representation learning method on hypergraph for 3D shape retrieval and recognition, called multi-scale hypergraph neural network (MHGNN). In this method, the correlation among 3D shapes is formulated in a hypergraph and a hypergraph convolution process is conducted to learn the representations. Here, multiple representations can be obtained through different convolution layers, leading to multi-scale representations of 3D shapes. A fusion module is then introduced to combine these representations for 3D shape retrieval and recognition. The main advantages of our method lie in 1) the high-order correlation among 3D shapes can be investigated in the framework and 2) the joint multi-scale representation can be more robust for comparison. Comparisons with state-of-the-art methods on the public ModelNet40 dataset demonstrate remarkable performance improvement of our proposed method on the 3D shape retrieval task. Meanwhile, experiments on recognition tasks also show better results of our proposed method, which indicate the superiority of our method on learning better representation for retrieval and recognition. [ABSTRACT FROM AUTHOR]