M\"obius 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, M\"obius-equivariant spherical convolution operator which we call M\"obius convolution, and with it, develop the foundations for M\"obius-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 M\"obius 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.
翻译:M\ “ obius” 转换在几何学和球形图像处理中都起着重要作用。 它们是由 2D 表面和同质体等同球体的相容自动形态构成的一组。 我们在这里展示了一本小说, M\ “ obius- equivalient 球形变异操作器 ” 我们称之为 M\ “ obius convolution, 并以此开发M\ “obius- equivariant 球形CNN ” 的基础。 我们的方法基于一个简单的观察: 要实现等同, 我们只需要考虑能够改变其周围框中所看到点位置的低维子分组。 要在比例上高效率地计算 M\ “ obius ” 和 obius convolutions 的变异作用, 我们就可以用快速的球形调变形和图像分割任务来计算光谱域中的变形。 由此产生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上取得有希望的结果来证明它的效用。