Anisotropic Spherical Scattering Networks via Directional Spin Wavelet

Scattering networks on Euclidean domains are capable of analytically realizing signal representation invariant to transformations such as translation, rotation and scaling with wavelets. However, existing scattering networks defined on the sphere and Riemannian manifolds only consider axisymmetric wavelets and are restricted in representation by the isotropic filter structures. In this paper, we propose a novel anisotropic spherical scattering network to achieve multi-scale directional representation for spherical signals. The scattering transform is realized by cascading directional spin wavelets and modulus operators to propagate localized signal components of varying scales and directions in a recursive manner without parameter tuning. Furthermore, a combined spherical scattering network is presented to guarantee the invariance to arbitrary rotations about the z-axis by incorporating the scattering coefficients along the dimension of rotation angle. To our best knowledge, the proposed network is the first to achieve multi-scale anisotropic filtering via the scattering transform on the sphere. We demonstrate in theory that the proposed network is energy preserving, invariant to azimuthal rotations and stable to diffeomorphisms. Extensive experimental results on benchmark datasets show that the proposed network achieves state-of-the-art performance in spherical signal analysis on various 2-D and 3-D datasets mapped to the sphere.