  Anisotropic diffusion processes with a diffusion tensor are important in
image analysis, physics, and engineering. However, their numerical
approximation has a strong impact on dissipative artefacts and deviations from
rotation invariance. In this work, we study a large family of finite difference
discretisations on a 3 x 3 stencil. We derive it by splitting 2-D anisotropic
diffusion into four 1-D diffusions. The resulting stencil class involves one
free parameter and covers a wide range of existing discretisations. It
comprises the full stencil family of Weickert et al. (2013) and shows that
their two parameters contain redundancy. Furthermore, we establish a bound on
the spectral norm of the matrix corresponding to the stencil. This gives time
step size limits that guarantee stability of an explicit scheme in the
Euclidean norm. Our directional splitting also allows a very natural
translation of the explicit scheme into ResNet blocks. Employing neural network
libraries enables simple and highly efficient parallel implementations on GPUs.
