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，其数值近似对耗散伪影和旋转不变性的偏差具有显著影响。本研究系统研究了3×3模板上的一大类有限差分离散格式。我们通过将二维各向异性扩散分解为四个一维扩散过程推导得出该格式。所得模板类包含一个自由参数，覆盖了现有多种离散格式的广泛范围。该模板类包含了Weickert等人(2013)的全模板族，并证明其双参数模型存在冗余性。此外，我们建立了该模板对应矩阵的谱范数界限，由此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条件。我们的方向分裂方法还可将显式格式自然地转化为ResNet模块。利用神经网络库，可在GPU上实现简单高效的并行计算。