Layered Fields for Natural Tessellations on Surfaces

TL;DR

This paper introduces a novel layered field model for natural surface tessellations that uses smooth functions and PDEs, avoiding complex geometric computations and enabling efficient parallel implementation.

Contribution

It proposes a new surface partitioning method based on layered fields and PDEs, simplifying computation and enhancing parallelization compared to traditional Voronoi-based approaches.

Findings

Avoids geodesic estimation and complex data structures

Uses simple linear algebra operations suitable for GPU acceleration

Enables parallel extraction of tessellations and dual meshes

Abstract

Mimicking natural tessellation patterns is a fascinating multi-disciplinary problem. Geometric methods aiming at reproducing such partitions on surface meshes are commonly based on the Voronoi model and its variants, and are often faced with challenging issues such as metric estimation, geometric, topological complications, and most critically parallelization. In this paper, we introduce an alternate model which may be of value for resolving these issues. We drop the assumption that regions need to be separated by lines. Instead, we regard region boundaries as narrow bands and we model the partition as a set of smooth functions layered over the surface. Given an initial set of seeds or regions, the partition emerges as the solution of a time dependent set of partial differential equations describing concurrently evolving fronts on the surface. Our solution does not require geodesic estimation, elaborate numerical solvers, or complicated bookkeeping data structures. The cost per time-iteration is dominated by the multiplication and addition of two sparse matrices. Extension of our approach in a Lloyd's algorithm fashion can be easily achieved and the extraction of the dual mesh can be conveniently preformed in parallel through matrix algebra. As our approach relies mainly on basic linear algebra kernels, it lends itself to efficient implementation on modern graphics hardware.