论文标题
Möbius卷积用于球形CNN
Möbius Convolutions for Spherical CNNs
论文作者
论文摘要
Möbius变换在几何形状和球形图像处理中都起着重要作用 - 它们是2D表面的共形自动形态和相当于同型的球形的组。在这里，我们介绍了一个小说，莫比乌斯 - 等级球形卷积操作员，我们称之为莫比乌斯卷积，并为此开发了Möbius-Equivariant球形CNN的基础。我们的方法基于一个简单的观察：为了实现均衡，我们只需要考虑较低维度的亚组，该亚组会改变邻居框架中所见点的位置。为了有效地计算Möbius卷积，我们得出了转换在球形过滤器上的作用的近似，使我们能够通过快速的球形谐波变换来计算光谱域中的卷积。最终的框架既灵活又具有描述性，我们通过在形状分类和图像分割任务中实现有希望的结果来证明其实用性。
Möbius transformations play an important role in both geometry and spherical image processing - they are the group of conformal automorphisms of 2D surfaces and the spherical equivalent of homographies. Here we present a novel, Möbius-equivariant spherical convolution operator which we call Möbius convolution, and with it, develop the foundations for Möbius-equivariant spherical CNNs. Our approach is based on a simple observation: to achieve equivariance, we only need to consider the lower-dimensional subgroup which transforms the positions of points as seen in the frames of their neighbors. To efficiently compute Möbius convolutions at scale we derive an approximation of the action of the transformations on spherical filters, allowing us to compute our convolutions in the spectral domain with the fast Spherical Harmonic Transform. The resulting framework is both flexible and descriptive, and we demonstrate its utility by achieving promising results in both shape classification and image segmentation tasks.
