The PR-star octree

Abstract

We propose the PR-star octree as a combined spatial data structure for performing efficient topological queries on tetrahedral meshes. The PR-star octree augments the Point Region octree (PR Octree) with a list of tetrahedra incident to its indexed vertices, i.e. those in the star of its vertices. Thus, each leaf node encodes the minimal amount of information necessary to locally reconstruct the topological connectivity of its indexed elements. This provides the flexibility to efficiently construct the optimal data structure to solve the task at hand using a fraction of the memory required for a corresponding data structure on the global tetrahedral mesh. Due to the spatial locality of successive queries in typical GIS applications, the construction costs of these runtime data structures are amortized over multiple accesses while processing each node. We demonstrate the advantages of the PR-star octree representation in several typical GIS applications, including detection of the domain boundaries, computation of local curvature estimates and mesh simplification.

A primal/dual representation for discrete Morse complexes on tetrahedral meshes

“…In [DFM10], a family of spatial indexes for tetrahedral meshes, called tetrahedral trees, has been introduced which generalizes similar data structures for maps and triangle meshes to 3D. A fundamentally and conceptually different data structure for tetrahedral meshes, called the PR-star octree [WFDV11], uses the spatial embedding of the mesh to index its topological connectivity (see Section 7). This data structure was used to extract morphological features from terrain and volume datasets in [DFIW12] using a streaming implementation of Robins et al's gradient computation.…”

“…We introduce a new encoding for discrete vector fields defined over irregular tetrahedral meshes in which information is attached only to the tetrahedra. Thus, it is well suited to be used in connection with any topological data structure which explicitly encodes only the vertices and the tetrahedra of the mesh [PBCF93, GR09,WFDV11]. We use this encoding as a compact representation for the discrete Morse gradient field of a discrete Morse function defined over the mesh.…”

“…In contrast to topological data structures, which explicitly encode the connectivity among mesh elements, or to spatial indexes, which index the elements for efficient location-based queries, a PR-star octree [WFDV11] uses the spatial index induced by an octree to efficiently generate local applicationdependent topological data structures at runtime. Here, we briefly describe the PR-star octree and discuss the extraction of the morphological features discussed in Section 6 on the PR-star octree.…”

“…We use a more compact representation for the leaves of the PR-star octree compared with the one in [WFDV11], by exploiting the spatial locality provided by the octree through a reindexing of arrays P and T . Besides the hierarchical information associated with the octree (e.g.…”

“…Here, we introduce a compact encoding for discrete vector fields which is only based on the tetrahedra, and is therefore suitable for combination with any such mesh representation. Moreover, we propose an implementation for the PR-star octree [WFDV11], which derives the local connectivity of the mesh through a spatial index on the mesh geometry. Thus, by trading a reasonable amount of computation at runtime, all connectivity relations can be compactly encoded.…”