Diamond-based models for scientific visualization

Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains including scientific visualization, finite element analysis and shape modeling and analysis. A popular class of such approaches is based on the regular simplex bisection operator, which bisects simplices (e.g. line segments, triangles, tetrahedra) along the midpoint of a predetermined edge. Regular simplex bisection produces adaptive simplicial meshes of high geometric quality, while simplifying the extraction of crack-free, or conforming, approximations to the original dataset. Efficient multiresolution representations for such models have been achieved in 2D and 3D by clustering sets of simplices sharing the same bisection edge into structures called diamonds. In this thesis, we introduce several diamond-based approaches for scientific visualization. We first formalize the notion of diamonds in arbitrary dimensions in terms of two related simplicial decompositions of hypercubes. This enables us to enumerate the vertices, simplices, parents and children of a diamond. In particular, we identify the number of simplices involved in conforming updates to be factorial in the dimension and group these into a linear number of subclusters of simplices that are generated simultaneously. The latter form the basis for a compact pointerless representation for conforming meshes generated by regular simplex bisection and for efficiently navigating the topological connectivity of these meshes. Secondly, we introduce the supercube as a high-level primitive on such nested meshes based on the atomic units within the underlying triangulation grid. We propose the use of supercubes to associate information with coherent subsets of the full hierarchy and demonstrate the effectiveness of such a representation for modeling multiresolution terrain and volumetric datasets. Next, we introduce Isodiamond Hierarchies, a general framework for spatial access structures on a hierarchy of diamonds that exploits the implicit hierarchical and geometric relationships of the diamond model. We use an isodiamond hierarchy to encode irregular updates to a multiresolution isosurface or interval volume in terms of regular updates to diamonds. Finally, we consider nested hypercubic meshes, such as quadtrees, octrees and their higher dimensional analogues, through the lens of diamond hierarchies. This allows us to determine the relationships involved in generating balanced hypercubic meshes and to propose a compact pointerless representation of such meshes. We also provide a local diamond-based triangulation algorithm to generate high-quality conforming simplicial meshes

Spatial Modeling using Triangular, Tetrahedral, and Pentatopic Decompositions

Techniques are described for facilitating operations for spatial modeling using triangular, tetrahedral, and pentatopic decompositions of the underlying domain. In the case of terrain data, techniques are presented for navigating between adjacent triangles of a hierarchical triangle mesh where the triangles are obtained by a recursive quadtree-like subdivision of the underlying space into four equilateral triangles. We describe a labeling technique for the triangles which is useful in implementing the quadtree triangle mesh as a linear quadtree (i.e., a pointer-less quadtree). The navigation can then take place in this linear quadtree. This results in algorithms that have a worst-case constant time complexity, as they make use of a fixed number of bit manipulation operations. In the case of volumetric data, we consider a multi-resolution representation based on a decomposition of a field domain into nested tetrahedral cells generated by recursive tetrahedron bisection, that we call a Hierarchy of Tetrahedra (HT). We describe our implementation of an HT, and discuss how to extract conforming meshes from an HT so as to avoid discontinuities in the approximation of the associated scalar field. This is accomplished by using worst-case constant time neighbor finding algorithms. We also present experimental results in connection with a set of basic queries for performing analysis of volume data sets at different levels of detail. In the case of four-dimensional data which can include time as the fourth dimension, we present a multi-resolution representation of a four-dimensional scalar field based on a recursive decomposition of a hypercubic domain into a hierarchy of nested four-dimensional simplexes, that we call a Hierarchy of Pentatopes (HP). This structure allows us to generate conforming meshes that avoid discontinuities in the corresponding approximation of the associated scalar field. Neighbor finding is an important part of this process and using our structure, it is possible to find neighbors in worst-case constant time by using bit manipulation operations, thereby avoiding traversing the hierarchy

Polyphospholyl ligands as building blocks for the formation of polymeric and spherical assemblies

This thesis is concerned with polyphospholyl ligands as building blocks for the formation of polymeric and spherical assemblies. Within the scope of this work, the coordination chemistry of 1,2,4-triphosphaferrocenes, 1,2,4-triphospholyl anions and pentaphosphaferrocenes towards Cu(I) halides is investigated. Thereby, the first two substance classes preferably tend to the build-up of novel one-, two- and three-dimensional polymers bearing rather uncommon or even unprecedented structural motifs. On the other hand, pentaphosphaferrocenes are capable of the template-directed synthesis of discrete, nano-sized supramolecules. Within a comprehensive study on the template requirements a series of small molecules could be encapsulated in these spheres for the first time. In addition, hitherto unknown host complexes were obtained which all differ in size, charge and topology. Besides the coordination chemistry, also novel complexes and salts were synthesized and characterized, which can now be used as building blocks in supramolecular chemistry