Realistic pattern formations on surfaces by adding arbitrary roughness

TL;DR

This work investigates how surface roughness affects pattern formation in reaction-diffusion systems by coupling RD dynamics to geometry through two rough-surface constructions: analytic parametric surfaces $\mathcal M$ built by random heightfields $z(x,y)$, and random rough surfaces $\mathcal S$ generated by heat-filter smoothing of random nodal data. An intrinsic finite-difference scheme is developed to solve the Laplace-Beltrami operator on $\mathcal M$, and RD patterns are simulated to study how roughness amplitude $\delta_{\mathcal M}$ and spatial frequencies $(M,N)$ deform spots and stripes, with results showing closer resemblance to animal coats as roughness increases. The paper additionally demonstrates qualitative agreement between patterns on $\mathcal M$ and $\mathcal S$, and provides a pathway to extend these methods to closed rough manifolds, enhancing realism in texture synthesis for biological and materials applications. Overall, the geometry-driven RD framework offers a principled approach to generating diverse, realistic surface patterns by controlling surface roughness and diffusion anisotropy.

Abstract

We are interested in generating surfaces with arbitrary roughness and forming patterns on the surfaces. Two methods are applied to construct rough surfaces. In the first method, some superposition of wave functions with random frequencies and angles of propagation are used to get periodic rough surfaces with analytic parametric equations. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in the generation of pattern formation. Numerical experiments show that the patterns become irregular as the amplitude and frequency of the rough surface increase. For the sake of easy generalization to closed manifolds, we propose a second construction method for rough surfaces, which uses random nodal values and discretized heat filters. We provide numerical evidence that both surface {construction methods} yield comparable patterns to those {observed} in real-life animals.

Figures (16)

• Figure 1: Real-life examples of skin patterns.
• Figure 1: A bird's-eye view depicting random rough surfaces ${\mathcal{M}}$ (left) by the construction method from Section \ref{['sec:introductionroughsurface']} and $\mathcal{S}$ (right) by the construction method from Section \ref{['sec:RRS']} .
• Figure 1: Accuracy and convergence results for the heat equation on the rough surface in Figure \ref{['fig:maxmineig']}(a) with $M=N=1, \delta_M=1E-2$: (a) Convergence with respect to spatial refinement. (b) Convergence with respect to time.
• Figure 1: Accuracy and convergence results for a reaction-diffusion system on the rough surface in Figure \ref{['fig:maxmineig']}(a) with $M=N=1,\delta_M=1E-2$. (a) Convergence with respect to spatial refinement. (b) Convergence with respect to time.
• Figure 1: Rough surfaces $\mathcal{S}$ with $\kappa=2$ and the number of filter steps $J=15$, subject to different filter-diffusion tensors $\mathcal{F}$.