We propose a definition for the spherical cross-correlation that is both expressive and rotation-equivariant. The spherical correlation satisfies a generalized Fourier theorem, which allows us to compute it efficiently using a generalized(non-commutative) Fast Fourier Transform (FFT) algorithm. We demonstrate the computational efficiency, numerical accuracy, and effectiveness of spherical CNNs applied to 3D model recognition and atomization energy regression.
? 关于SO3 作为刚体变换的阐述，参考：半闲居士视觉SLAM十四讲笔记(3)三维空间刚体运动 - par..._CSDN博客 。
? ? 区分出三维图像和平面的细微差别，把球面图像看做是三维流形，把球面展开为离散的三维李群，把SO(3)的关系用CNNs的高层进行表示。
? ? As shown in Figure 1, there is no good way to use translational convolution or cross-correlation1 to analyze spherical signals. The most obvious approach, then, is to change the definition of crosscorrelation by replacing filter translations by rotations. Doing so, we run into a subtle but important difference between the plane and the sphere: whereas the space of moves for the plane (2D translations) is itself isomorphic to the plane, the space of moves? for the sphere (3D rotations) is a different, three-dimensional manifold called SO(3)2. It follows that the result of a spherical correlation (the output feature map) is to be considered a signal on SO(3), not a signal on the sphere, S2. For this reason, we deploy SO(3) group correlation in the higher layers of a spherical CNN (Cohen and Welling, 2016).
? ? The implementation of a spherical CNN (S2-CNN) involves two major challenges. Whereas a square grid of pixels has discrete translation symmetries, no perfectly symmetrical grids for the sphere exist. This means that there is no simple way to define the rotation of a spherical filter by one pixel. Instead, in order to rotate a filter we would need to perform some kind of interpolation. The other challenge is computational efficiency; SO(3) is a three-dimensional manifold, so a naive implementation of SO(3) correlation is O(n6).