map channel). Note that the input solid textures include surface textures as well as interior textures. We present a method for representing solid objects with spatiallyvarying oriented textures by repeatedly pasting solid texture exemplars. The underlying concept is to extend the 2D texture patchpasting approach of lapped textures to 3D solids using a tetrahedral mesh and 3D texture patches. The system places texture patches according to the user-defined volumetric tensor fields over the mesh to represent oriented textures. We have also extended the original technique to handle nonhomogeneous textures for creating solid models whose textural patterns change gradually along the depth fields. We identify several texture types considering the amount of anisotropy and spatial variation and provide a tailored user interface for each. With our simple framework, large-scale realistic solid models can be created easily with little memory and computational cost. We demonstrate the effectiveness of our approach with several examples including trees, fruits, and vegetables.

Summary. The recent Sparse Voronoi Refinement (SVR) Algorithm for mesh generation has the fastest theoretical bounds for runtime and memory usage. We present a robust practical software implementation of the SVR for meshing a piecewise linear complex in 3 dimensions. Our software is competitive in runtime with state of the art freely available packages on generic inputs, and on pathological worse cases inputs, we show SVR indeed leverages its theoretical guarantees to produce vastly superior runtime and memory usage. The theoretical algorithm description of SVR leaves open several data structure design options, especially with regard to point location strategies. We show that proper strategic choices can greatly effect constant factors involved in runtime. 1

Abstract—Marching Cubes is a popular choice for isosurface extraction from regular grids due to its simplicity, robustness, and efficiency. One of the key shortcomings of this approach is the quality of the resulting meshes, which tend to have many poorly shaped and degenerate triangles. This issue is often addressed through postprocessing operations such as smoothing. As we demonstrate in experiments with several data sets, while these improve the mesh, they do not remove all degeneracies and incur an increased and unbounded error between the resulting mesh and the original isosurface. Rather than modifying the resulting mesh, we propose a method to modify the grid on which Marching Cubes operates. This modification greatly increases the quality of the extracted mesh. In our experiments, our method did not create a single degenerate triangle, unlike any other method we experimented with. Our method incurs minimal computational overhead, requiring at most twice the execution time of the original Marching Cubes algorithm in our experiments. Most importantly, it can be readily integrated in existing Marching Cubes implementations and is orthogonal to many Marching Cubes enhancements (particularly, performance enhancements such as out-of-core and acceleration structures). Index Terms—Meshing, marching cubes. Ç 1

Summary. A number of approaches have been suggested for the selection of the positions of Steiner points in Delaunay mesh refinement. In particular, one can define an entire region (called picking region or selection disk) inside the circumscribed sphere of a poor quality element such that any point can be chosen for insertion from this region. The two main results which accompany most of the point selection schemes, including those based on regions, are the proof of termination of the algorithm and the proof of good gradation of the elements in the final mesh. In this paper we show that in order to satisfy only the termination requirement, one can use larger selection disks and benefit from the additional flexibility in choosing the Steiner points. However, if one needs to keep the theoretical guarantees on good grading then the size of the selection disk needs to be smaller. We introduce two types of selection disks to satisfy each of these two goals and prove the corresponding results on termination and good grading first in two dimensions and then in three dimensions using the radius-edge ratio as a measure of element quality. We call the point placement method semi-generalized because the selection disks are defined only for mesh entities of the highest dimension (triangles in two dimensions and tetrahedra in three dimensions); we plan to extend these ideas to lower-dimensional entities in the future work. We implemented the use of both two- and three-dimensional selection disks into the available Delaunay refinement libraries and present one example (out of many choices) of a point placement method; to the best of our knowledge, this is the first implementation of Delaunay refinement with point insertion at any point of the selection disks (picking regions). 1

Summary. We propose a new mesh refinement algorithm for computing quality guaranteed Delaunay triangulations in three dimensions. The refinement relies on new ideas for computing the goodness of the mesh, and a sampling strategy that employs numerically stable Steiner points. We show through experiments that the new algorithm results in sparse well-spaced point sets which in turn leads to tetrahedral meshes with fewer elements than the traditional refinement methods.

Mesh refinement is the problem to produce a triangulation (typically Delaunay) of an input set of points augmented by Steiner points, such that every triangle or tetrahedron has good quality (no small angles). The requirement arises from the applications: in scientific computing and in graphics, meshes are often used to discretely represent the value of a function over space. In addition to the quality requirement, the user often has input segments or polygons (generally, a piecewise linear complex) they would like see retained in the mesh; the mesh must respect these constraints. Finally, the mesh should be size-conforming: the size of mesh elements should be related to a particular sizing function based on the distance between input features. The static meshing problem is increasingly well-understood: one can download software with provable guarantees that on reasonable input, the meshes will have good quality, will respect the input, and will be size-conforming; more recently, these algorithms have started to come with optimal runtimes of O(n lg(L/s) + m), where L/s is the spread of the input. As a first result, I present experimental results of the first time-optimal code, available online at sparse-meshing.com. Increasingly, static meshing is insufficient: users want to modify the mesh over time. Throwing away the old mesh and remeshing from scratch is a common approach, but that suffers from slow runtime, and from reinterpolation error because the old and new meshes may be almost unrelated. Mesh stability analyzes the correspondence between meshes for two inputs. The main theoretical result of this thesis is an algorithm that has provable bounds on stability: upon inserting or removing a feature that in the final mesh is represented using k points, the mesh only modifies O(k lg(L/s)) mesh simplices. Finally, stability can be exploited to produce an efficient dynamic algorithm. Under the self-adjusting computation framework, with a small amount of additional effort, I show that my algorithm can be dynamized to run in O(k lg(L/s)) time per update, using O(n lg(L/s) + m) space.

In this paper we introduce the constrained tetrahedralization as a new acceleration structure for ray tracing. A constrained tetrahedralization of a scene is a tetrahedralization that respects the faces of the scene geometry. The closest intersection of a ray with a scene is found by traversing this tetrahedralization along the ray, one tetrahedron at a time. We show that constrained tetrahedralizations are a viable alternative to current acceleration structures, and that they have a number of unique properties that set them apart from other acceleration structures: constrained tetrahedralizations are not hierarchical yet adaptive; the complexity of traversing them is a function of local geometric complexity rather than global geometric complexity; constrained tetrahedralizations support deforming geometry without any effort; and they have the potential to unify several data structures currently used in global illumination.

In this paper we introduce the constrained tetrahedralization as a new acceleration structure for ray tracing. A constrained tetrahedralization of a scene is a tetrahedralization that respects the faces of the scene geometry. The closest intersection of a ray with a scene is found by traversing this tetrahedralization along the ray, one tetrahedron at a time. We show that constrained tetrahedralizations are a viable alternative to current acceleration structures, and that they have a number of unique properties that set them apart from other acceleration structures: constrained tetrahedralizations are not hierarchical yet adaptive; the complexity of traversing them is a function of local geometric complexity rather than global geometric complexity; constrained tetrahedralizations support deforming geometry without any effort; and they have the potential to unify several data structures currently used in global illumination.