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Anisotropic Diffusion Stencils: From Simple Derivations over Stability Estimates to ResNet Implementations

Karl Schrader, Joachim Weickert, Michael Krause

TL;DR

Anisotropic diffusion with diffusion tensor $\bm D$ poses numerical challenges due to dissipative artefacts and rotation invariance. The authors propose the $\delta$-stencil, a $3\times3$ finite-difference discretisation derived by directional splitting into four 1-D diffusions with a single free parameter $\delta$, unifying and simplifying the Weickert et al. stencil family. They establish a fairly tight spectral-norm bound on the associated matrix, yielding explicit time-step limits for stable explicit schemes, e.g., $\tau \le 2/\rho(\bm A)$, and discuss behavior in homogeneous and maximally anisotropic cases. They further translate the discretisation into a ResNet-style architecture, enabling GPU-accelerated implementations via PyTorch and highlighting cross-fertilisation between numerical PDEs and neural networks.

Abstract

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.

Anisotropic Diffusion Stencils: From Simple Derivations over Stability Estimates to ResNet Implementations

TL;DR

Anisotropic diffusion with diffusion tensor poses numerical challenges due to dissipative artefacts and rotation invariance. The authors propose the -stencil, a finite-difference discretisation derived by directional splitting into four 1-D diffusions with a single free parameter , unifying and simplifying the Weickert et al. stencil family. They establish a fairly tight spectral-norm bound on the associated matrix, yielding explicit time-step limits for stable explicit schemes, e.g., , and discuss behavior in homogeneous and maximally anisotropic cases. They further translate the discretisation into a ResNet-style architecture, enabling GPU-accelerated implementations via PyTorch and highlighting cross-fertilisation between numerical PDEs and neural networks.

Abstract

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.
Paper Structure (5 sections, 2 theorems, 26 equations, 1 figure)

This paper contains 5 sections, 2 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Discretising Anisotropic Diffusion with the $\bm \delta$-Stencil
  3. Stability Theory for the $\bm\delta$-Stencil
  4. Translating Anisotropic Diffusion into ResNets
  5. Conclusions

Key Result

theorem thmcountertheorem

Let the eigenvalues of $\bm D$ be given by $\lambda_1 \ge \lambda_2 \ge 0$. Assume that $\,\delta = \alpha(a\!+\!c) + \beta b\,$ where $\,\beta=\gamma(1\!-\!2\alpha)\,\textup{sgn}(b)\,$ for $\alpha \in [0, \frac{1}{2}]$ and $|\gamma|\leq 1$. Then the spectral norm of the matrix $\bm A$ satisfies

Figures (1)

  • Figure 1: (a) Left: Translation of 1-D nonlinear diffusion into a ResNet block. Adapted from ASAPW21. (b) Right: Anisotropic diffusion as a ResNet block with a sum of four 1-D divergence blocks. The blocks $\bm{B}_0,...,\bm{B}_3$ correspond to the directions $\bm{e}_0,...,\bm{e}_3$.

Theorems & Definitions (3)

  • theorem thmcountertheorem: Bound on Spectral Norm
  • proof
  • corollary thmcountercorollary: Stability of Explicit Scheme