SGMNet: Learning Rotation-Invariant Point Cloud Representations via Sorted Gram Matrix

Recently, various works that attempted to introduce rotation invariance to point cloud analysis have devised point-pair features, such as angles and distances. In these methods, however, the point-pair is only comprised of the center point and its adjacent points in a vicinity, which may bring information loss to the local feature representation. In this paper, we instead connect each point densely with all other points in a local neighborhood to compose the point-pairs. Specifically, we present a simple but effective local feature representation, called sorted Gram matrix(SGM), which is not only invariant to arbitrary rotations, but also models the pair-wise relationship of all the points in a neighbor-hood. In more detail, we utilize vector inner product to model distance- and angle-information between two points, and in a local patch it naturally forms a Gram matrix. In order to guarantee permutation invariance, we sort the correlation value in Gram matrix for each point, therefore this geometric feature names sorted Gram matrix. Furthermore, we mathematically prove that the Gram matrix is rotation-invariant and sufficient to model the inherent structure of a point cloud patch. We then use SGM as features in convolution, which can be readily integrated as a drop-in module into any point-based networks. Finally, we evaluated the proposed method on two widely used datasets, and it outperforms previous state-of-the-arts on both shape classification and part segmentation tasks by a large margin.