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.

翻译：具有扩散张量的各向异性扩散过程在图像分析、物理学和工程学中具有重要意义。然而，其数值近似会对耗散伪影和旋转不变性偏差产生显著影响。本文研究了一个基于3×3模板的有限差分离散化方法族。我们通过将二维各向异性扩散分解为四个一维扩散过程来推导该模板。所得模板类别包含一个自由参数，覆盖了现有离散化方法的广泛范围。该模板类涵盖了Weickert等人(2013)提出的完整模板族，并表明其两个参数存在冗余性。此外，我们建立了该模板对应矩阵的谱范数边界，从而给出了保证显式格式在欧几里得范数下稳定性的时间步长限制。我们的方向分裂方法还使得显式格式能够非常自然地转化为ResNet模块。利用神经网络库可以在GPU上实现简单且高效的并行计算。