A graph-theoretical approach for the computation of connected iso-surfaces based on volumetric data (1402.3236v1)

Abstract: The existing combinatorial methods for iso-surface computation are efficient for pure visualization purposes, but it is known that the resulting iso-surfaces can have holes, and topological problems like missing or wrong connectivity can appear. To avoid such problems, we introduce a graph-theoretical method for the computation of iso-surfaces on cuboid meshes in $\mathbb{R}^3$. The method for the generation of iso-surfaces employs labeled cuboid graphs $G(V,E,\mathcal{F})$ such that $V$ is the set of vertices of a cuboid $C\subset\mathbb{R}^3$, $E$ is the set of edges of $C$ and $\mathcal{F}\,:\,V\rightarrow [0,1]$. The nodes of $G$ are weighted by the values of $\mathcal{F}$ which represents the volumetric information, e.g.\ from a Volume of Fluid method. Using a given iso-level $c\in (0,1)$, we first obtain all iso-points, i.e.\ points where the value $c$ is attained by the edge-interpolated $\mathcal{F}$-field. The iso-surface is then built from iso-elements which are composed of triangles and are such that their polygonal boundary has only iso-points as vertices. All vertices lie on the faces of a single mesh cell. We give a proof that the generated iso-surface is connected up to the boundary of the domain and it can be decomposed into different oriented components. Two different components may have discrete points or line segments in common. The graph-theoretical method for the computation of iso-surfaces developed in this paper enables to recover local information of the iso-surface that can be used e.g.\ to compute discrete mean curvature and to solve surface PDEs. Concerning the computational effort, the resulting algorithm is as efficient as existing combinatorial methods.