On the minimum number of simplex shapes in longest edge bisection refinement of a regular n-simplex

In several areas like Global Optimization using branch-and-bound methods, the unit n-simplex is refined by bisecting the longest edge such that a binary search tree appears. This process generates simplices belonging to different shape classes. Having less simplex shapes facilitates the prediction of the further workload from a node in the binary tree, because the same shape leads to the same sub-tree. Irregular sub-simplices generated in the refinement process may have more than one longest edge when n\geqslant 3. The question is how to choose the longest edge to be bisected such that the number of shape classes is as small as possible. We develop a Branch-and-Bound (B&B) algorithm to find the minimum number of classes in the refinement process. The developed B&B algorithm provides a minimum number of eight classes for a regular 3-simplex. Due to the high computational cost of solving this combinatorial problem, future research focuses on using high performance computing to derive the minimum number of shapes in higher dimensions

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

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

A Geometric Toolbox for Tetrahedral Finite Element Partitions

In this work we present a survey of some geometric results on tetrahedral partitions and their refinements in a unified manner. They can be used for mesh generation and adaptivity in practical calculations by the finite element method (FEM), and also in theoretical finite element (FE) analysis. Special emphasis is laid on the correspondence between relevant results and terminology used in FE computations, and those established in the area of discrete and computational geometry (DCG)

Multi-Level Shape Representation Using Global Deformations and Locally Adaptive Finite Elements

We present a model-based method for the multi-level shape, pose estimation and abstraction of an object’s surface from range data. The surface shape is estimated based on the parameters of a superquadric that is subjected to global deformations (tapering and bending) and a varying number of levels of local deformations. Local deformations are implemented using locally adaptive finite elements whose shape functions are piecewise cubic functions with C1 continuity. The surface pose is estimated based on the model\u27s translational and rotational degrees of freedom. The algorithm first does a coarse fit, solving for a first approximation to the translation, rotation and global deformation parameters and then does several passes of mesh refinement, by locally subdividing triangles based on the distance between the given datapoints and the model. The adaptive finite element algorithm ensures that during subdivision the desirable finite element mesh generation properties of conformity, non-degeneracy and smoothness are maintained. Each pass of the algorithm uses physics-based modeling techniques to iteratively adjust the global and local parameters of the model in response to forces that are computed from approximation errors between the model and the data. We present results demonstrating the multi-level shape representation for both sparse and dense range data

Spatial Decompositions for Geometric Interpolation and Efficient Rendering

Interpolation is fundamental in many applications that are based on multidimensional scalar or vector fields. In such applications, it is possible to sample points from the field, for example, through the numerical solution of some mathematical model. Because point sampling may be computationally intensive, it is desirable to store samples in a data structure and estimate the values of the field at intermediate points through interpolation. We present methods based on building dynamic spatial data structures in which the samples are computed on-demand, and adaptive strategies are used to avoid oversampling. We first show how to apply this approach to accelerate realistic rendering through ray-tracing. Ray-tracing can be formulated as a sampling and reconstruction problem, where rays in 3-space are modeled as points in a 4-dimensional parameter space. Sample rays are associated with various geometric attributes, which are then used in rendering. We collect and store a relatively sparse set of sampled rays, and use inexpensive interpolation methods to approximate the attribute values for other rays. We present two data structures: (1) the <i>ray interpolant tree (RI-tree)</i>, which is based on a kd-tree-like subdivision of space, and (2) the <i>simplex decomposition tree (SD-tree)</i>, which is based on a hierarchical regular simplicial mesh, and improves the functionality of the RI-tree by guaranteeing continuity. For compact storage as well as efficient neighbor computation in the mesh, we present a pointerless representation of the SD-tree. An essential element of this approach is the development of a location code that enables efficient access and navigation of the data structure. For this purpose we introduce a location code, called an LPTcode, that uniquely encodes the geometry of each simplex of the hierarchy. We present rules to compute the neighbors of a given simplex efficiently through the use of this code. We show how to traverse the associated tree and how to answer point location and interpolation queries. Our algorithms work in arbitrary dimensions. We also demonstrate the use of the SD-tree for rendering atmospheric effects. We present empirical evidence that our methods can produce renderings of good quality significantly faster than simple ray-tracing