Delaunay-restricted Optimal Triangulation of 3D Polygons

Triangulation of 3D polygons is a well studied topic of research. Existing methods for finding triangulations that minimize given metrics (e.g., sum of triangle areas or dihedral angles) run in a costly O(n4) time [BS95,BDE96], while the triangulations are not guaranteed to be free of intersections. To address these limitations, we restrict our search to the space of triangles in the Delaunay tetrahedralization of the polygon. The restriction allows us to reduce the running time down to O(n2) in practice (O(n3) worst case) while guaranteeing that the solutions are intersection free. We demonstrate experimentally that the reduced search space is not overly restricted. In particular, triangulations restricted to this space usually exist for practical inputs, and the optimal triangulation in this space approximates well the optimal triangulation of the polygon. This makes our algorithms a practical solution when working with real world data

Intersurf: dynamic interface between proteins

Protein docking is a fundamental biological process that links two proteins. This link is typically defined by an interaction between two large zones of the protein boundaries. Visualizing such an interface is useful to understand the process thanks to 3D protein structures, to estimate the quality of docking simulation results, and to classify interactions in order to predict docking affinity between classes of interacting zones. Since the interface may be defined by a surface that separates the two proteins, it is possible to create a map of interaction that allows comparisons to be performed in 2D. This paper presents a very fast algorithm that extracts an interface surface and creates a valid and low-distorted interaction map. Another benefit of our approach is that a pre-computed part of the algorithm enables the surface to be updated in real-time while residues are moved

Feature-sensitive and Adaptive Image Triangulation: A Super-pixel-based Scheme for Image Segmentation and Mesh Generation

With increasing utilization of various imaging techniques (such as CT, MRI and PET) in medical fields, it is often in great need to computationally extract the boundaries of objects of interest, a process commonly known as image segmentation. While numerous approaches have been proposed in literature on automatic/semi-automatic image segmentation, most of these approaches are based on image pixels. The number of pixels in an image can be huge, especially for 3D imaging volumes, which renders the pixel-based image segmentation process inevitably slow. On the other hand, 3D mesh generation from imaging data has become important not only for visualization and quantification but more critically for finite element based numerical simulation. Traditionally image-based mesh generation follows such a procedure as: (1) image boundary segmentation, (2) surface mesh generation from segmented boundaries, and (3) volumetric (e.g., tetrahedral) mesh generation from surface meshes. These three majors steps have been commonly treated as separate algorithms/steps and hence image information, once segmented, is not considered any more in mesh generation. In this thesis, we investigate a super-pixel based scheme that integrates both image segmentation and mesh generation into a single method, making mesh generation truly an image-incorporated approach. Our method, called image content-aware mesh generation, consists of several main steps. First, we generate a set of feature-sensitive, and adaptively distributed points from 2D grayscale images or 3D volumes. A novel image edge enhancement method via randomized shortest paths is introduced to be an optional choice to generate the features’ boundary map in mesh node generation step. Second, a Delaunay-triangulation generator (2D) or tetrahedral mesh generator (3D) is then utilized to generate a 2D triangulation or 3D tetrahedral mesh. The generated triangulation (or tetrahedralization) provides an adaptive partitioning of a given image (or volume). Each cluster of pixels within a triangle (or voxels within a tetrahedron) is called a super-pixel, which forms one of the nodes of a graph and adjacent super-pixels give an edge of the graph. A graph-cut method is then applied to the graph to define the boundary between two subsets of the graph, resulting in good boundary segmentations with high quality meshes. Thanks to the significantly reduced number of elements (super-pixels) as compared to that of pixels in an image, the super-pixel based segmentation method has tremendously improved the segmentation speed, making it feasible for real-time feature detection. In addition, the incorporation of image segmentation into mesh generation makes the generated mesh well adapted to image features, a desired property known as feature-preserving mesh generation

Homeomorphic Tetrahedralization of Multi-material Images with Quality and Fidelity Guarantees

We present a novel algorithm for generating three-dimensional unstructured tetrahedral meshes of multi-material images. The algorithm produces meshes with high quality since it provides a guaranteed dihedral angle bound of up to 19.47° for the output tetrahedra. In addition, it allows for user-specified guaranteed bounds on the two-sided Hausdorff distance between the boundaries of the mesh and the boundaries of the materials. Moreover, the mesh boundary is proved to be homeomorphic to the object surface. The algorithm is fast and robust, it produces a sufficiently small number of mesh elements that comply with these guarantees, as compared to other software. The theory and effectiveness of our method are illustrated with the experimental evaluation on synthetic and real medical data

Deformable Simplicial Complexes

In this dissertation we present a novel method for deformable interface tracking in 2D and 3D|deformable simplicial complexes (DSC). Deformable interfaces are used in several applications, such as fluid simulation, image analysis, reconstruction or structural optimization. In the DSC method, the interface (curve in 2D; surface in 3D) is represented explicitly as a piecewise linear curve or surface. However, the domain is also subject to discretization: triangulation in 2D; tetrahedralization in 3D. This way, the interface can be alternatively represented as a set of edges/triangles separating triangles/tetrahedra marked as outside from those marked as inside. Such an approach allows for robust topological adaptivity. Among other advantages of the deformable simplicial complexes there are: space adaptivity, ability to handle and preserve sharp features, possibility for topology control. We demonstrate those strengths in several applications. In particular, a novel, DSC-based fluid dynamics solver has been developed during the PhD project. A special feature of this solver is that due to the fact that DSC maintains an explicit interface representation, surface tension is more easily dealt with. One particular advantage of DSC is the fact that as an alternative to topology adaptivity, topology control is also possible. This is exploited in the construction of cut loci on tori where a front expands from a single point on a torus and stops when it self-intersects

Doctor of Philosophy

dissertationOne of the fundamental building blocks of many computational sciences is the construction and use of a discretized, geometric representation of a problem domain, often referred to as a mesh. Such a discretization enables an otherwise complex domain to be represented simply, and computation to be performed over that domain with a finite number of basis elements. As mesh generation techniques have become more sophisticated over the years, focus has largely shifted to quality mesh generation techniques that guarantee or empirically generate numerically well-behaved elements. In this dissertation, the two complementary meshing subproblems of vertex placement and element creation are analyzed, both separately and together. First, a dynamic particle system achieves adaptivity over domains by inferring feature size through a new information passing algorithm. Second, a new tetrahedral algorithm is constructed that carefully combines lattice-based stenciling and mesh warping to produce guaranteed quality meshes on multimaterial volumetric domains. Finally, the ideas of lattice cleaving and dynamic particle systems are merged into a unified framework for producing guaranteed quality, unstructured and adaptive meshing of multimaterial volumetric domains