SOT

Abstract

The Corner Table (CT) promoted by Rossignac et al. provides a simple and efficient representation of triangle meshes, storing 6 integer references per triangle (3 vertex references in the V table and 3 references to opposite corners in the O table that accelerate access to adjacent triangles). The Compact Half Face (CHF) proposed by Lage et al. extends CT to tetrahedral meshes, storing 8 references per tetrahedron (4 in the V table and 4 in the O table). We call it the Vertex Opposite Table (VOT) and propose a sorted variation, SVOT, which does not require any additional storage and yet provides, for each vertex, a reference to an incident corner from which an incident tetrahedron may be recovered and the star of the vertex may be traversed at a constant cost per visited element. We use a set of powerful wedge-based operators for querying and traversing the mesh. Finally, inspired by tetrahedral mesh encoding techniques used by Weiler et al. and by Szymczak and Rossignac, we propose our Sorted O Table (SOT) variation, which eliminates the V table completely and hence reduces storage requirements by 50% to only 4 references and 9 bits per tetrahedron, while preserving the vertex-to-incident-corner references and supporting our wedge operators with a linear average cost.Several tetrahedral mesh compression schemes have been proposed [31,61]. Some support progressive refinements [45] or streaming [8, 33, 66]. Unfortunately, the compressed format they offer is not suitable for traversing, simplifying [14,22,21,67], refining [41], or improving [57, 42, 58] the mesh. Thus, an effective representation scheme is needed that provides efficient support for random access operators that traverse the mesh and which may be constructed efficiently from other (possibly compressed) formats or updated to reflect mesh modifications.

Triangulation Data Structures

“…The main idea used in the SOT data structure [9] is to implicitly represent the map from triangles to corners (triangle operator), and the map from corners to vertices (vertex operator), thourough face reordering. First match each vertex to an incident triangle (in such a way a triangle is matched with at most one vertex).…”

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

“…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.…”

“…When encoding unstructured tetrahedral meshes, the IG can be verbose, since it explicitly encodes all vertices, edges, faces and tetrahedra in the mesh plus several topological connectivity relations. On the contrary, data structures which encode only the vertices and the tetrahedra [PBCF93,GR09] have been shown to be much more compact [CDW11]. 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.…”