Simplex and Diamond Hierarchies: Models and Applications

Abstract

Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multiresolution models of scalar fields, such as terrains, and static or time-varying volume data. They have also been used as an alternative to quadtrees and octrees as spatial access structures and in other applications. In this state of the art report, we distinguish between approaches that focus on a specific dimension and those that apply to all dimensions. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplexes, called diamonds, as the modeling primitive. This leads to two classes of data structures and to different query approaches. We present the hierarchical models in a dimension-independent manner, and organize the description of the various applications, primarily interactive terrain rendering and isosurface extraction, according to the dimension of the domain.

Bisection-Based Triangulations of Nested Hypercubic Meshes

“…In dimension d, this generates at most d similarity classes of simplices [13] that we refer to as RSB simplices. A recent survey on the RSB scheme and its applications can be found in [31]. …”

“…Σ is a simplicial complex defined by a collection of Regular Simplex Bisection (RSB) simplices. The properties of RSB simplices [13,31] ensure the quality of the triangulation. Specifically, the simplices in the mesh belong to at most d similarity classes of well-shaped simplices, and the valence of a vertex is at most 2 d d!.…”

“…Specifically, we propose a simplicial decomposition for nested hypercubic meshes generated by applying a regular simplex bisection rule to locally triangulate each hypercube in the mesh. Due to the properties of this bisection scheme [13,25,4,29,31], our algorithm generates adaptive simplicial complexes composed of high quality simplices from a small set of geometric primitives. Furthermore, compatibility between adjacent triangulated hypercubes is implicitly enforced and the complexity of the mesh increases by a dimension-dependent multiplicative constant.…”

A Comparison of Milp and Minlp Solver Perfomance on the Example of a Drinking Water Supply System Design Problem

“…An aggregated convex combination formulation based on a Freudenthal triangulation [9] is used as piecewise linear approximation method. The surfaces which describe the pump curves ∆H(n, Q) and P (n, Q) are cut into triangles.…”

Coarse-grained multiresolution structures for mobile exploration of gigantic surface models

“…Each diamond is composed by the set of all the tetrahedra sharing their longest edge. Using a diamond based structure, see Weiss and De Floriani [2010], dependencies are implicitly encoded into the hierarchy, and refinement is interruptible, producing a conforming mesh also when children data is not available. This feature perfectly fits in a mobile rendering architecture, where it is common to experience data fetch delays due to bandwidth limitations.…”