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Paper

#### Layered Fields for Natural Tessellations on Surfaces

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##### Fulltext (public)

arXiv:1804.09152.pdf

(Preprint), 6MB

##### Supplementary Material (public)

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##### Citation

Zayer, R., Mlakar, D., Steinberger, M., & Seidel, H.-P. (2018). Layered Fields for Natural Tessellations on Surfaces. Retrieved from http://arxiv.org/abs/1804.09152.

Cite as: http://hdl.handle.net/21.11116/0000-0002-152D-5

##### 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.