  Convolutional Neural Networks (CNNs) have become the method of choice for
learning problems involving 2D planar images. However, a number of problems of
recent interest have created a demand for models that can analyze spherical
images. Examples include omnidirectional vision for drones, robots, and
autonomous cars, molecular regression problems, and global weather and climate
modelling. A naive application of convolutional networks to a planar projection
of the spherical signal is destined to fail, because the space-varying
distortions introduced by such a projection will make translational weight
sharing ineffective.
  In this paper we introduce the building blocks for constructing spherical
CNNs. 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.
