@article {13328,
	title = {Simplex and Diamond Hierarchies: Models and Applications},
	journal = {Computer Graphics Forum},
	volume = {30},
	year = {2011},
	month = {2011/11/24/},
	pages = {2127 - 2155},
	abstract = {Hierarchical spatial decompositions are a basic modelling tool in a variety of application domains. Several papers on this subject deal with hierarchical simplicial decompositions generated through regular simplex bisection. Such decompositions, originally developed for finite elements, are extensively used as the basis for multi-resolution 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. The primary distinction among all such approaches is whether they treat the simplex or clusters of simplices, called diamonds, as the modelling 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.},
	keywords = {hierarchy of diamonds, hierarchy of simplices, I.3.5 [Computer Graphics]: Computational Geometry and Object Modelling{\textemdash} Hierarchy and geometric transformations, I.3.6 [Computer Graphics]: Methodology and Techniques{\textemdash}Graphics data structures and data types, interactive terrain visualization, mesh-based multiresolution models, multiresolution isosurfaces, nested refinement schemes, Regular simplex bisection, scalar field visualization, spatial access structures},
	isbn = {1467-8659},
	doi = {10.1111/j.1467-8659.2011.01853.x},
	url = {http://onlinelibrary.wiley.com/doi/10.1111/j.1467-8659.2011.01853.x/abstract?userIsAuthenticated=false\&deniedAccessCustomisedMessage=},
	author = {Weiss,K. and De Floriani, Leila}
}
@article {13287,
	title = {Diamond Hierarchies of Arbitrary Dimension},
	journal = {Computer Graphics Forum},
	volume = {28},
	year = {2009},
	month = {2009/08/31/},
	pages = {1289 - 1300},
	abstract = {Nested simplicial meshes generated by the simplicial bisection decomposition proposed by Maubach [Mau95] have been widely used in 2D and 3D as multi-resolution models of terrains and three-dimensional scalar fields, They are an alternative to octree representation since they allow generating crack-free representations of the underlying field. On the other hand, this method generates conforming meshes only when all simplices sharing the bisection edge are subdivided concurrently. Thus, efficient representations have been proposed in 2D and 3D based on a clustering of the simplices sharing a common longest edge in what is called a diamond. These representations exploit the regularity of the vertex distribution and the diamond structure to yield an implicit encoding of the hierarchical and geometric relationships among the triangles and tetrahedra, respectively. Here, we analyze properties of d-dimensional diamonds to better understand the hierarchical and geometric relationships among the simplices generated by Maubach{\textquoteright}s bisection scheme and derive closed-form equations for the number of vertices, simplices, parents and children of each type of diamond. We exploit these properties to yield an implicit pointerless representation for d-dimensional diamonds and reduce the number of required neighbor-finding accesses from O(d!) to O(d).},
	keywords = {I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling{\textemdash}Hierarchy and geometric transformations, I.3.6 [Computer Graphics]: Methodology and Techniques{\textemdash}Graphics data structures and data types},
	isbn = {1467-8659},
	doi = {10.1111/j.1467-8659.2009.01506.x},
	url = {http://onlinelibrary.wiley.com/doi/10.1111/j.1467-8659.2009.01506.x/abstract?userIsAuthenticated=false\&deniedAccessCustomisedMessage=},
	author = {Weiss,Kenneth and De Floriani, Leila}
}
